SYLLABUS  OF  MATHEMATICS 


A  SYMPOSIUM  COMPILED  BY  THE  COM- 
MITTEE ON  THE  TEACHING  OF  MATHE- 
MATICS TO  STUDENTS   OF  ENGINEERING 


Accepted  by  the 

Society  for  the  Promotion  of  Engineering  Education 

AT  THE  Nineteenth  Annual  Meeting  held  at 

Pittsburgh,  Pa.,  June,  1911 


Office  of  the  Secretary 

ITHACA,  N.  Y. 

1912 


Press  oh 

The  new  era  Printing  company 

Lancaster,  pa. 


TABLE   OF   CONTENTS. 


PAGE 

Keport  of  the  Committee  on  the  Teaching  of  Mathematics  to 

Students  of  Engineering  1 

Syllabus  of  Elementary  Algebra 5 

Chap.      I.     Transformation  of  Algebraic  Expressions 6 

Chap.     II.     Solution  of  Equations  13 

Chap.  III.     Miscellaneous  Topics   16 

Syllabus  of  Elementary  Geometry  and  Mensuration 19 

Syllabus  of  Plane  Trigonometry 27 

Chap.      I.     Sine,  Cosine  and  Tangent  of  Acute  Angles  .....  28 

Chap.     II.     The  Trigonometric  Funcdons  of  any  Angle 33 

Chap.  III.     General  Propertiesof  the  Trigonometric  Functions  43 

Syllabus  of  Analytic  Geometry 47 

Chap.         I.     Eectangular  Coordinates 48 

Chap.        II.     The  Straight  Line 50 

Chap.      III.     The  Circle  52 

Chap.      IV.     The  Parabola   53 

Chap.        V.     The  Ellipse 57 

Chap.      VI.     The  Hyperbola 61 

Chap.    VII.     Transformation  of  Coordinates  65 

Chap.  VIII.     General  Equations  of  the  Second  Degree  in  x 

and  y  66 

Chap.      IX.     Systems  of  Conies 70 

Chap.        X.     Polar  Coordinates  71 

Chap.       XI.     Coordinates  in  Space   72 

Syllabus  of  Differential  and  Integral  Calculus 75 

Chap.       I.     Functions  and  Their  Graphical  Representation .. .  77 
Chap.     II.     Differentiation.     Rate  of  Change  of  a  Function  82 
Chap.  III.     Integration  as  Anti-Differentiation.    Simple  Dif- 
ferential Equations 96 

Chap.    IV.     Integration  as   the   Limit   of   a   Sum.     Definite 

Integrals    102 

Chap.      V.     Applications  to  Algebra.     Expansion  in  Series; 

Indeterminate  Forms   109 

Chap.    VI.     Applications  to  Geometry  and  Mechanics 114 

Discussion  at  the  Pittsburgh  Meeting 119 

Syllabus  on  Complex  Quantities 134 


267886 


REPORT  OF  THE  COMMITTEE  ON  THE  TEACH- 
ING OF  MATHEMATICS  TO  STUDENTS 
OF  ENGINEERING. 

To  the  Society  for  the  Promotion  of  Engineering  Education: 
The  committee  was  appointed  at  a  joint  meeting  of  mathe- 
maticians and  engineers  held  in  Chicago,  December  30-31, 
1907,  under  the  auspices  of  the  Chicago  Section  of  the  Ameri- 
can Mathematical  Society,  and  Sections  A  and  D  of  the 
American  Association  for  the  Advancement  of  Science,*  and 
on  the  suggestion  of  officers  of  the  Society  for  the  Promotion 
of  Engineering  Education  who  were  there  present,  the  com- 
mittee was  instructed  to  report  to  this  Society. 
The  membership  of  the  committee  is  as  follows : 

Alger,  Philip  R.,  professor  of  mathematics,  U.  S.  Navy, 
Annapolis,  Md. 

Campbell,  Donald  F.,  professor  of  mathematics.  Armour 
Institute  of  Technology,  Chicago,  111. 

Engler,  Edmund  A.,  president  of  the  Worcester  Polytechnic 
Institute,  Worcester,  Mass. 

Haskins,  Charles  N.,  assistant  professor  of  mathematics,  Dart- 
mouth College,  Hanover,  N.  H. 

Howe,  Charles  S.,  president,  Case  School  of  Applied  Science, 
Cleveland,  Ohio. 

KuiCHLiNG,  Emil,  consulting  civil  engineer.  New  York  City. 

Magruder,  William  T.,  professor  of  mechanical  engineering, 
Ohio  State  University,  Columbus,  Ohio. 

MoDJESKi,  Ralph,  civil  engineer,  Chicago,  111. 

Osgood,  William  F.,  professor  of  mathematics.  Harvard  Uni- 
versity, Cambridge,  Mass. 

Slighter,  Charles  S.,  consulting  engineer  of  the  U.  S.  Recla- 
mation Service,  professor  of  applied  mathematics.  Univer- 
sity of  Wisconsin,  Madison,  Wis. 

*For  an  account  of  the  Chicago  meeting,  see  Science  for  1908  (July 
12,  24,  and  31;  August  7  and  28;  and  September  4). 

1 


Z  COMMITTEE  ON  TEACHING  MATHEMATICS. 

Steinmetz,  Charles  P.,  consulting  engineer  of  the  General 
Electric    Company,    professor    of    electrical    engineering, 
Union  University,  Schenectady,  N.  Y. 
Swain,   George  F.,   consulting  engineer,   professor   of  civil 

engineering,  Harvard  University,  Cambridge,  Mass. 
TowNSEND,  Edgar  J.,  dean  of  the  College  of  Science  and  pro- 
fessor of  mathematics.  University  of  Illinois,  Urbana,  111. 
TuRNEAURE,  Frederick  E.,  dean  of  the  College  of  Mechanics 
and  Engineering,  University  of  Wisconsin,  Madison,  Wis. 
Waldo,  Clarence  A.,  head  professor  of  mathematics,  Washing- 
ton University,  St.  Louis,  Mo. 
Williams,  Gardner  S.,  consulting  engineer,  professor  of  civil, 
hydraulic  and  sanitary  engineering.  University  of  Michigan, 
Ann  Arbor,  Mich. 
Woodward,  Calvin  M.,  dean  of  the  School  of  Engineering  and 
Architecture  and  professor  of  mathematics  and  applied 
mechanics,  Washington  University,  St.  Louis,  Mo. 
Woodward,  Robert  S.,  president  of  the  Carnegie  Institution  of 

Washington,  Washington,  D.  C. 
ZiWET,  Alexander,  professor  of  mathematics,  University  of 

Michigan,  Ann  Arbor,  Mich. 
Huntington,  Edward  V.,  chairman,  assistant  professor  of 
mathematics.  Harvard  University,  Cambridge,  Mass. 
After  deliberation,  the  committee  decided  that  it  could  best 
carry  out  the  purpose  for  which  it  was  appointed  by  preparing 
a  synopsis  of  those  fundamental  principles  and  methods  of 
mathematics  which,  in  the  opinion  of  the  committee,  should 
constitute  the  minimum  mathematical  equipment  of  the  stu- 
dent of  engineering. 

This  synopsis,  as  finally  adopted,  consists  of  five  parts : 

1.  A  Syllabus  of  Elementary  Algebra; 

2.  A  Syllabus  of  Elementary  Geometry  and  Mensuration; 

3.  A  Syllabus  of  Plane  Trigonometry ; 

4.  A  Syllabus  of  Analytic  Geometry; 

5.  A  Syllabus  of  Differential  and  Integral  Calculus. 

Two  other  syllabi,  on  Numerical  Computation  and  on  Ele- 
mentary Dynamics,  were  contemplated  in  the  original  plan, 
but  were  not  completed. 


COMMITTEE  ON  TEACHING  MATHEMATICS.  3 

It  is  hoped  that  this  report  may  be  serviceable  in  two  ways : 
first,  to  the  teacher,  as  an  indication  of  where  the  emphasis 
should  be  laid;  and  secondly,  to  the  student,  as  a  syllabus  of 
facts  and  methods  which  are  to  be  his  working  tools.  It  does 
not  include  data  for  which  the  student  would  properly  refer 
to  an  engineers'  hand-hook;  it  includes  rather  just  those 
things  for  which  he  ought  never  to  he  ohliged  to  refer  to  any 
hook — the  things  which  he  should  have  constantly  at  his 
fingers'  ends. 

The  teacher  of  mathematics  should  see  to  it  that  at  least 
these  facts  are  perfectly  familiar  to  all  his  students,  so  that 
the  professor  of  engineering  may  presuppose,  with  confidence, 
at  least  this  much  mathematical  knowledge  on  the  part  of  his 
students.  On  the  other  hand,  if  the  professor  of  engineering 
needs  to  use,  at  any  point,  more  advanced  mathematical  meth- 
ods than  those  here  mentioned,  he  should  be  careful  to  explain 
them  to  his  class. 

The  committee  has  not  found  it  possible  to  propose  a  de- 
tailed course  of  study.  The  order  in  which  these  topics 
should  be  taken  up  must  be  left  largely  to  the  discretion  of 
the  individual  teacher.  The  committee  is  firmly  of  the 
opinion,  however,  that  whatever  order  is  adopted,  the  principal 
part  of  the  course  should  be  prohlems  worked  hy  the  students, 
and  that  all  these  problems  should  be  solved  on  the  basis  of  a 
small  number  of  fundamental  principles  and  methods,  such 
as  are  here  suggested. 

The  defects  in  the  mathematical  training  of  the  student  of 
engineering  appear  to  be  largely  in  knowledge  and  grasp  of 
fundamental  principles,  and  the  constant  effort  of  the  teacher 
should  be  to  ground  the  student  thoroughly  in  these  funda- 
mentals, which  are  too  often  lost  sight  of  in  a  mass  of  details. 

A  pressing  need  at  the  present  time  is  a  series  of  synoptical 
text-books,  which  shall  present,  (1)  the  fundamental  prin- 
ciples of  the  science  in  compact  form,  and  (2)  a  classified  and 
graded  collection  of  problems  (which  would  naturally  be  sub- 
ject to  continual  change  and  expansion) .  It  is  the  hope  of  the 
committee  that  this  report,  which  is  confined  to  the  first  part  of 


4  COMMITTEE    ON    TEACHING   MATHEMATICS. 

the  desired  text-book,  will  stimulate  throughout  the  country 
practical  contributions  toward  the  second. 

In  the  early  part  of  its  investigation  the  committee  collected 
a  large  amount  of  information  in  regard  to  the  present  status 
of  mathematical  instruction  for  engineering  students.  Since 
that  time,  however,  a  much  more  inclusive  inquiry  has  been 
undertaken  by  the  International  Commission  on  the  Teaching 
of  Mathematics,  of  which  the  American  Commissioners  are 
Professors  D.  E.  Smith,  J.  W.  A.  Young  and  W.  F.  Osgood. 
In  order  to  avoid  unnecessary  duplication,  this  committee 
voted  to  turn  over  all  the  results  of  its  own  inquiry  in  this  field 
to  the  larger  commission,  to  be  worked  up  in  accordance  with 
the  general  scheme  adopted  by  that  commission,  and  to  be 
incorporated  in  their  report.  This  material  is  therefore  not 
included  in  the  present  report. 

KespectfuUy  submitted, 

Edward  Y.  Huntington, 
Chairman, 

June,  1911. 


A  SYLLABUS  OF  THE  FORMAL  PART  OF 
ELEMENTARY  ALGEBRA. 

This  syllabus  is  intended  to  include  those  facts  and  methods  of  ele- 
mentary algebra  which  a  student  who  has  completed  a  course  in  that 
subject  should  be  expected  to  **know  by  heart" — that  is,  those  funda- 
mental principles  which  he  ought  to  have  made  so  completely  a  perma- 
nent part  of  his  mental  equipment  that  he  will  never  need  to  *  *  look  them 
up  in  a  book." 

It  is  not  intended  as  a  program  of  study  for  beginners,  and  no  at- 
tempt has  been  made  to  arrange  the  topics  in  the  order  in  which  they 
riiould  be  taught.  In  reviewing  the  subject,  however,  either  at  the  end 
of  the  course  in  algebra,  or  at  the  beginning  of  any  later  course,  such  a 
syllabus  will  be  found  serviceable  to  both  teacher  and  student;  and  in 
the  hands  of  a  skillful  teacher,  and  supplemented  by  an  adequate  collec- 
tion of  problems,  it  might  well  be  made  the  basis  of  a  course  of  study 
conducted  by  the  "syllabus  method." 

One  of  the  chief  defects  in  the  present-day  teaching  of  algebra  is  the 
multiplicity  of  detached  rules  with  which  the  student 's  mind  is  burdened;* 
and  every  successful  attempt  to  knit  together  a  number  of  these  detached 
rules  into  a  single  general  principle  (provided  this  principle  is  simple 
and  easily  applied)  should  conduce  to  economy  of  mental  effort,  and  di- 
minish the  liability  to  error. 

Table  op  Contents. 

Chapter  I.    Transformation  op  Algebraic  Expressions. 

General  laws  of  addition  and  multiplication. 

Type-forms  of  multiplication  (Factoring). 

Fractions. 

Negatives. 

Eadicals  and  Imaginaries. 

Exponents  and  Logarithms. 
Chapter  II.    Solution  op  Equations. 

Legitimate  operations  on  equations. 

To  solve  a  single  equation. 
Quadratic  equations. 
Exponential  equations. 

To  solve  a  set  of  simultaneous  equations. 
Chapter  III.    Miscellaneous  Topics. 

Katio  and  proportion. 

Variation. 

Inequalities. 

Arithmetical,  geometric,  and  harmonic  progressions. 

*  For  example,  in  a  recent  prominent  text-book  there  are  no  less  than 
■fifty  italicized  rules  in  the  part  of  the  book  preceding  quadratic 
equations  I  * 


CHAPTER  I. 

Transformation  of  Algebraic  Expressions. 

1.  The  ordinary  operations  of  transforming  and  simplify- 
ing algebraic  expressions  should  be  so  familiar  to  the  student 
that  he  performs  them  almost  instinctively ;  at  the  same  time 
he  should  be  able,  whenever  called  upon,  to  justify  each  step  of 
his  work  by  reference  to  some  one  or  more  of  a  small  number 
of  well  established  principles. 

For  example,  if  the  student  is  asked  6t/  what  authority  lie  replaces 
r-j^  by  ^,  or  Va'  +  5'  hj  a -{- h  (to  mention  only  two  of  the  common- 
est blunders),  he  will  be  forced  to  recognize  either  that  he  is  making  use 
of  methods  that  he  has  never  proved,  and  that  are  in  fact  erroneous, 
or  else  (which  is  more  likely)  that  he  is  working  altogether  in  the  dark, 
without  any  conscious  reason  for  the  steps  he  has  taken. 

The  following  list  of  such  principles,  while  making  no  pre- 
tense at  logical  completeness,  will  be  sufficient  for  all  practical 
purposes. 

2.  General  laws  of  addition  and  multiplication. 

a-\-h  =  h  -{-a.  ah  =  })a,  ( Commutative  laws. ) 

(a  +  h)+c  =  a+{b  +  c).  {ah)c  =  a{hc). 

(Associative  laws.) 

a{h  -\-  c)  =  ah  -\-  ac.  (Distributive  law. ) 

a  +  0  =  a.  aXl  =  «.  aXO  =  0. 

These  laws  hold  when  a,  h,  c  are  any  of  the  quantities  that  occur  in 
ordinary  algebra,  whether  ''real'*  or  ** complex. '' *  The  student  should 
be  constantly  encouraged  to  test  general  algebraic  statements  by  substi- 
tuting concrete  numerical  values. 

3.  Type-forms  of  multiplication  (Factoring). 

The  following  type-forms  of  multiplication  are  the  ones  that  are  most 
important  to  remember: 

•  This  syllabus  is  confined  chiefly  to  the  algebra  of  real  quantities;  the 
algebra  of  complex  quantities  will  be  treated  only  incidentally. 


ALGEBRA.  / 

a«  — &«=(«  —  &)  (a2  +  a&  +  &2), 
and  so  on;  the  general  case  is  best  remembered  in  the  form 

l^x^=il  —  x){l  +  x  +  x^  +  x^-\ [-x^-'). 

Note  also  that  in  the  algebra  of  real  quantities,  a»  +  &"  is 
divisible  by  a  +  &  when  and  only  when  n  is  odd.    Thus : 

a^-]-h^=={a  +  h){a^  —  al)  +  h^). 

Further:  {x  + a)  (x  +  h)  =x^  +  {a  +  l)x  +  a'b, 

and  the  "binomial  theorem": 

{a  +  &)2  =  a2  _|_  2ah  +  h\     {a  +  hy  =  a'  +  da'h  +  Sah^  +  &^ 

{a  —  h)'  =  a^  —  2al)  +  h\    {a  —  by  =  a^  —  da'h  +  Sah^  —  h^; 

{a  +  h)-  = 

3  !  -^  I 

where  A;!  =  *'A;  f actorial' '  =  1  X  2  X  3  X  •••  X^;. 

4.  Fractions. 

a 
Def .    If  &ic  =  a,  then  and  only  then  we  write  x  =  t  (or  a/b, 

OT  a-^b). 

Here  a  is  called  the  numerator  and  6  the  denominator  of  the  fraction. 
A  fraction  with  a  zero  denominator,  as  a/0,  does  not  represent  any 
definite  quantity.  For,  if  o  is  not  zero,  there  is  no  quantity  x  such  that 
0  X  so  =  a;  and  if  o  =  0,  then  every  quantity  x  will  have  this  property. 
Hence,  the  denominator  of  a  fraction  must  always  be  different  from  zero. 

From  the  definition,  a/1  =  a ;  also 

a  0 

-  =  1,     -  =  0,  (a  4=  0).* 

*  The  symbol  =|=  means  *  *  not  equal  to. ' ' 


O  '  ALGEBEA. 

To  add  two  fractions  with  common  denominator : 

a      b      a  -\-  b 
c        c~~      G      ' 
To  multiply  two  fractions : 

a      X      ax 
b      y"  by' 
To  divide  by  a  fraction,  ''invert  the  divisor  and  multiply*': 
a      X      a      y      ay 
T)  '  y'~b      x^  bx' 

The  value  of  a  fraction  is  not  changed  if  we  multiply  {or 
divide)  both  the  numerator  and  the  denominator  by  any 
quantity  not  zero: 

a      ma 

l  =  ^b  (™  +  o). 

This  is  the  most  important  principle  concerning  fractions. 

For  example,  to  reduce  two  fractions  to  a  common  denominator,  we 
have  merely  to  multiply  numerator  and  denominator  of  each  fraction  by 
a  suitable  factor. 

Again,  to  simplify  a  complex  fraction,  we  multiply  the  whole  numera- 
tor and  the  whole  denominator  by  any  quantity  which  will  *' absorb'*  all 
the  wbsidiary  denominators.     Thus,  by  multiplying  by  xyz,  we  have 

£+* 

X        y am  +  hxz 

c  -\-  d        (c^-  rf)xy ' 

2 

at  once,  by  a  single  mental  process.  (The  common  practice  of  reducing 
the  numerator  and  denominator  separately,  and  then  inverting  the  denom- 
inator and  multiplying,  is  tedious  and  clumsy.) 

Def .  If  bx  =^  1,  then  x  =  1/b,  which  is  called  the  reciprocal 
of  &.  To  divide  by  &  (&  =t=  0)  is  the  same  as  to  multiply  by  the 
reciprocal  of  &. 

5.  Negatives. 

Def.  If  fl  +  ic  =  0,  then  and  only  then  we  write  x= — a. 
In  particular,  —  ( — a)  =a. 

If  a  is  not  zero,  —  a  is  always  opposite  to  a ;  that  is,  if  a  is 
positive,  —  a  is  negative,  and  if  a  is  negative,  —  a  is  positive. 


ALGEBEA.  9 

Thus,  if  a  =  —  3,  which  is  a  negative  quantity,  thcr.  —  o  =  3,  which 
is  positive. 

The  notation  |  a  |,  which  is  coming  into  use  more  and  more 
widely,  means  the  absolute  value  of  a,  that  is,  the  numerical 
value  of  a  regardless  of  sign ;  thus,  |  5  |  =  5,  |  —  5  |  =  5. 

The  laws  of  operation  with  the  minus  sign  are  best  remem- 
bered by  regarding  —  a  as  the  product  of  a  and  — 1 : 

{—l)Xa=—a, 

whence,  in  particular  (putting  a  =  —  1), 

(-1)X(-1)=1. 
When  this  is  done,  the  customary  formulas: 

{—a){—b)=ab,     {—a){b)=  —  ab,     —  =r:-^=r  — ^,    :^j^  =  p 

become  immediate  consequences  of  the  general  laws  of  multiplication 
and  division,  and  therefore  need  not  be  separately  memorized;  and  the 
same  is  true  of  the  formula 

a  —  (x  +  y  —  g)=a  —  x  —  y-{-z, 

which,  when  remembered  in  the  following  form,  becomes  an  immediate 
application  of  the  distributive  law :  *  *  a  minus  sign  in  front  of  a  paren- 
thesis must  be  *  distributed '  through  every  term  within,  if  the  parenthe- 
ses are  to  be  taken  away.'' 

By  knitting  together  in  this  way  the  rules  for  negatives  with  the 
general  rules  of  operation,  the  total  number  of  processes  to  be  remem- 
bered and  applied,  and  hence  the  liability  to  error,  is  materially  reduced. 

Def .  It  a-\-x  =  b,  then  and  only  then  we  write  x  =  b  —  a. 
It  is  easily  shown  that  b  —  a  =  b  -\-  ( —  a) ;  that  is,  subtracting 
any  quantity  a  is  the  same  as  adding  the  opposite  of  a, 

6.  Radicals. 

Def.  If  a  is  positive,  and  n  is  any  positive  integer,  there 
will  always  be  one  positive  value  of  x  such  that  x*  =  a.  This 
value  X  is  denoted  by  ^/a,  and  is  called  the  (principal)  nth 
root  of  a. 

It  should  be  noticed  that  while  there  are  (for  example)  two  square 
roots  of  9,  namely  3  and  —  3,  it  is  only  the  positive  one  of  these  two 
values  that  is  denoted  by  V9;  that  is,  the  mark  VS  means  3  and  not  —  3. 


10  ALGEBRA. 

The  radical  sign,  except  in  the  case  of  square  roots,  and 
sometimes  in  the  case  of  cube  roots,  should  always  be  replaced 
by  fractional  exponents  (see  below)  when  it  is  desired  to  com- 
pute with  these  quantities;  this  done,  no  special  rules  for  the 
manipulation  of  radicals  need  then  be  remembered  beyond  the 
general  laws  of  exponents. 

Square  roots.    If  a  and  h  are  positive, 

•\/Wb  =  ayh,    and     ya^J'b=^^/db. 

Note  also  the  process  called  **  rationalizing  the  denominator 
(or  numerator)  of  a  fraction";  for  example, 

c  c  Va  — V6      c(Va  ■~Af6) 

Va-fV6       Va+V6       Va  —  V6  a  —  b 

Vi— aj       Vl— a;       Vl— a;        l—a; 
X 


Vl  -f  a;       Vl  +  aj       Vl  —  a;       Vl  —  aj^ 

Def .  If  a  is  negative,  and  n  is  odd,  there  will  always  be  one 
negative  value  of  x  such  that  x^  =  a\  this  value  is  denoted  by 

•^a,  and  is  called  the  (principal)  nth  root  of  a. 

Thus  ^—^z=  —  2. 

7.  Imaginaries. 

If  a  is  negative,  and  n  is  even,  then  there  is  no  positive  or  negative 
nth  root  of  a.  Hence,  such  quantities  do  not  occur  in  the  algebra  of 
positive  and  negative  quantities.  They  occur  only  in  the  more  general 
algebra  of  complex  quantities;  in  this  algebra  every  quantity  a  (except 
zero)  has  n  distinct  nth  roots,  the  notation  '^o  being  applied,  as  occasion 
requires,  to  any  one  of  these  n  values.  The  detailed  study  of  this  general 
algebra  is  probably  too  difB.cult  for  a  first  course;  for  such  applications 
as  occur  in  elementary  work,  the  following  working  rules  are  sufficient : 


1)  In  manipulating  a  complex  quantity  of  the  form  V — b, 


where  h  is  positive,  write  V —  ^  =  V —  1  yb  =  iyb,  and  treat 
%  like  any  other  letter;  then  simplify  the 'result  by  the  relation 
t2  =  — 1. 

2)  Every  complex  quantity  can  be  written  in  the  form 
a  +  t6,  where  a  and  &  are  "rear*  (that  is,  positive,  negative, 
or  zero) ;  and  if  a  +  *&  =  o'  +  *^'»  then  a  =  a'  and  h  =  h\ 


ALGEBRA.  11 

In  electrical  engineering  the  letter  t  is  used  to  denote  current,  and 
V —  1  is  denoted  by  j. 

8.  Exponents. 

The  subject  of  negative  and  fractional  exponents  is  a  part  of  algebra 
in  which  the  preparation  of  the  student  is  apt  to  be  especially  unsatis- 
factory. 

Definition  of  negative  and  fractional  exponents.  If  a  is  positive,  and 
p  and  q  are  any  positive  integers,  then 

a<>  — 1,         a^=— ,         a^'^=i^a,        avli—\/aP. 

9.  Laws  of  operation  with  exponents. 
If  a  and  &  are  positive,  then ; 

«"=!,       «""=i^'        «""  =  > 

All  these  laws  hold  for  any  values  of  m  and  n;  the  three  fundamental 
ones  can  readily  be  recalled  to  mind  through  simple  special  cases,  such  as 
a*a\  (a»)»,  and  (o6)». 

The  three  other  laws  commonly  mentioned,  namely 

am-n=:^am/anj    am/n^:  *^^^     (a/b)m=zam/1)% 

are  immediate  corollaries  of  those  just  mentioned. 

If  a  is  negative,  and  m  not  an  integer,  a"*  will,  in  general,  be  a  complex 
quantity.  In  such  cases,  let  a'  =  —  a,  so  that  a'  is  positive,  and  write 
o"»=  ( — !)♦»«'»«,  where  ( — 1)"»  must  then  be  handled  according  to  the 
rules  of  operation  in  the  algebra  of  complex  quantities. 

10.  Logarithms. 

The  subject  of  logarithms  should  be  taught  in  logical  connection 
with  the  subject  of  exponents.  The  common  practice  of  separating  these 
subjects,  and  treating  logarithms  as  a  part  of  trigonometry,  is  unfortu- 
nate. Numerous  applications  of  logarithms  can  be  found  that  have 
nothing  to  do  with  trigonometry;  moreover,  the  training  in  the  use  of 
logarithms  which  a  student  gets  in  trigonometry  is  usually  quite  inade- 
quate as  a  preparation  for  the  applications  of  logarithms  in  any  of  his 
later  work  outside  of  surveying. 

Def.  The  logarithm  of  a  (positive)  number,  to  any  (posi- 
tive) base,  is  the  exponent  of  the  power  to  which  the  base  must 
be  raised  to  produce  that  number. 


12  ALOEBBA. 

Thus,  the  notation 


means 


log,N 


l'  =  N, 


Note  that  negative  numbers  in  general  have  no  logarithms  in  the 
algebra  of  real  quantities. 

From  the  laws  of  exponents  we  have,  whatever  the  base 
may  be : 

log  (a&)  =^log  a  +  log  &,    log  ^   =log  a  — log  6, 

log  (a**)  =n  log  a,  log  V«=  -log a, 

log  1  =  0,    log(6a5e)=l. 

Only  two  bases  are  in  common  use.  For  purposes  of 
numerical  computation,  the  base  chosen  is  10,  and  in  this 

system  log(10»)^n. 

In  higher  mathematics,  the  base  e  =  2.718  •  •  •  is  used,  for  the 
reason  that  the  use  of  this  base  simplifies  certain  formulas  in 
the  calculus;  in  this  system  log  {e^)=n. 

Change  of  hose.  To  find  logeN  when  logio^V"  is  known,  let 
a;  =  logeiV,  that  is,  e^  =  N,  Then  take  the  logarithm  of 
both  sides  of  this  equation  to  base  10,  and  solve  for  x. 

The  resulting  formula,  logg^=  (logioN')/(logioC),  is  so  easily  obtained 
in  this  way  that  it  is  not  worth  while  to  remember  it  separately.  The 
approximate  values 

logio<?  =  .4343,    and    log«A^=  (2.3026)   log^^, 

however,  are  useful  to  remember. 


CHAPTER  II. 

Solution  of  Equations. 

11.  Legitimate  operations  on  equations.  If  a  given  equa- 
tion is  true,  it  will  still  be  true  if  we 

(a)  add  any  quantity  we  please  to  both  sides; 

(&)  subtract  any  quantity  we  please  from  both  sides; 

(c)  multiply  both  sides  by  any  quantity  we  please; 

(d)  divide  both  sides  by  any  quantity  we  please  except  zero, 

(e)  raise  both  sides  to  any  positive  integral  power; 

(/)  *extract  any  positive  integral  root  of  both  sides,  except 
that  if  an  even  root  is  extracted,  the  double  sign  dz  must  be 
used; 

{g)  *take  the  logarithm  of  both  sides  (provided  both  sides 
are  positive). 

In  regard  to  (d),  we  must  never  divide  both  sides  by  an  unknown 
quantity  without  first  excluding  the  possibility  that  that  quantity  is  zero. 

In  (/),  the  restriction  stated  means,  for  example,  that  from  A*  =  B 
we  can  infer  merely  that  -4  =  ±  V5;  that  is,  that  either  A=iy/B,  or 
A  =  —  V-B;  but  we  cannot  tell  which. 

12.  To  solve  a  single  equation  in  x,  means  to  find  all  the 
values  of  x  that  satisfy  the  equation,  or  to  show  that  none 
such  exist. 

Any  value  of  x  that  satisfies  the  equation  is  called  a  root  of 
the  equation. 

In  testing  a  root,  the  only  safe  method  is  to  substitute  the  given 
value  in  each  side  of  the  equation  separately,  and  see  whether  the  re- 
sults, when  reduced,  are  equal.  Thus,  we  should  find  that  x=.  —  2  is  a 
root  of  the  equation  x  =  2  —  V12  —  2a;,  and  that  a;  =  4  is  not  a  root. 

In  this  connection  it  should  be  noticed  that  if  we  square  both  sides 
of  a  given  equation,  the  new  equation  will,  in  general,  have  more  roots 
than  th€  given  equation.  Thus  (to  use  the  same  example),  by  squaring 
X  —  2  =  —  V12  —  2a;  we  have  x*  —  2ar  —  8  =  0.  This  equation  has  of 
course  the  root  — 2,  since  x  =  —  2  satisfies  the  original  equation  from 

*  In  the  algebra  of  complex  quantities  (/)  and  {g)  are  not  applicable. 
2  13 


14  ALGEBRA. 

which  this  was  derived;  but  it  has  also  the  root  4,  which  was  not  a  root 
of  the  original  equation. 

The  formal  process  usually  called  *' solving  the  equation** 
means  merely  transforming  the  equation,  by  a  judicious  choice 
of  the  legitimate  operations,  into  a  form  in  which  the  solutions 
are  obvious. 

If  this  is  not  possible,  we  must  have  recourse  to  the  method 
of  trial  and  error  which,  while  often  laborious,  is  always 
applicable  in  numerical  cases. 

If  an  equation  is  given  in  the  factored  form : 

(x  —  a)(x  —  p){x  —  y)  ...=0, 

then  the  roots  are  obviously  x=:a,  x  =  p,  a;  =  7, . .  •  .    Thus,  the  roots 
of  x(x  +  2)  =  0  are  0  and  —  2. 

13.  Quadratic  equations.    To  solve  the  quadratic  equation 
ax^  +  hx  +  c  =  0, 
we  may  divide  through  by  a: 

x^  -{--X  = 

a  a 

and  then  ** complete  the  square**: 


whence, 


,6         /by        b'       c 
—  6  db  V6^  —  4ac 


Aac 


4a 


2      t 


05  = 


2a 


or,  we  may  use  the  general  result  just  obtained  as  a  formula. 

The  quantity  which  must  be  added  to  both  sides  in  "completing  the 

square  *  *  is  obvious  by  analogy  with  ic*  +  2mx  +  m',  so  that  this  method 

^  requires  less  effort  of  the  memory  than  the  method  of  solution  by  formula. 

The  *  *  method  of  factoring ' '  is  very  convenient  in  certain  special  cases, 
when  the  factors  can  be  obtained  by  inspection. 

The  method  still  sometimes  used,  of  first  multiplying  through  by  4o  to 
avoid  fractions,  is  apt  to  lead  to  confusion,  and  should  be  discouraged. 

From  the  formula  it  is  evident  that  the  sum  of  the  roots  is 


ALGEBBA.  15 

^1  +  ^2  =  —  ^A>  ^^^  the  product  of  the  roots  is  x^x^=^cla\ 
also,  if  the  coefficients,  a,  &,  c,  are  real,  the  roots  will  be  real- 
and-distinct,  real-and-coincident,  or  imaginary,  according  as 
\^*— 4ad:  is  positive,  zero,  or  negative. 

^^4.  Exponential  equations.  To  solve  an  equation  of  the 
form  a^=b,  when  a  and  h  are  positive,  take  the  logarithm  of 
both  sides:  x  log  a=log  &;  and  then  solve  for  x. 

15.  To  solve  a  set  of  simultaneous  equation  in  x,  y,  z  •  •  • 
means  to  find  all  the  sets  of  values  oi  x,  y,  z,  •  •  • ,  that  satisfy 
all  the  equations  at  once,  or  show  that  none  such  exist. 

Two  simultaneous  equations  of  the  first  degree,  as  oa?  +  6y  =  c  and 
Ax  -\-  By  =  C,  can  always  be  solved  in  a  couple  of  lines,  if  the  work  is 
arranged  as  follows: 


/ 


7x—   6y=-l  |_5 

14x  —  lOy  =  3  Is 


—  2 
1 


(— 354-42)x^  — 5  +  9  v 

(12  — 10)y  =  — 2  +  3 

whence  the  values  of  x  and  y  are  obvious,  provided  aB  —  hA  is  not  zero. 
(If  aB  —  6^  =  0,  there  is  either  no  pair  of  values  x,  y  that  satisfies  both 
the  equations,  or  else  there  are  an  infinite  number  of  pairs  of  values  that 
do  so ;  in  this  latter  case,  the  equations  are  not  independent,  that  is,  either 
of  them  can  be  derived  from  the  other.) 

The  theory  of  simultaneous  equations,  and  sometimes  the  numerical 
computation,  is  facilitated  by  the  use  of  determinants. 

In  general,  n  independent  equations  will  suffice  to  deter- 
mine in  unknown  quantities. 


CHAPTER  III. 

Miscellaneous  Topics. 

16.  Ratio  and  Proportion.    The  **  ratio  of  a 
simply  the  fraction  a/b;  and  a  "proportion'*  is  simply  an 
equation  between  two  ratios. 

The  notation  a'.h'.'.cld  should  be  replaced  by  the  equation  a/h=:c/d; 
and  all  special  terminology,  such  as  **  taking  a  proportion  by  alterna- 
tion/' '*by  composition/'  etc.,  should  be  dropped  in  favor  of  the 
ordinary  language  of  the  equation. 

17.  Variation.  The  statement  ''y  varies  as  x/'  or  ''2/  varies 
directly  as  x/'  or  **!/  is  proportional  to  x/'  means  y  =  kXf 
where  k  is  some  constant.  Similarly,  ^'y  varies  inversely  as 
ic,"  means  y  =  k/x;  **y  varies  inversely  as  the  square  of  x," 
means  y  =  k/x^.  The  constant  k  can  always  be  determined  if 
we  know  any  pair  of  values  of  x  and  y  that  belong  together. 

The  statement  *  *  y  varies  as  u  and  v, ' '  means  y  varies  as  the  product  of 
u  and  V,  that  is,  y=z1cuv. 

18.  Inequalities.  The  notions  of  ' '  greater  and  less ' '  are  thoroughly 
familiar  when  we  are  dealing  only  with  positive  quantities,  but  the  ex- 
tension of  these  terms  to  the  algebra  of  all  real  quantities  (positive, 
negative,  and  zero)  is  apt  to  cause  some  confusion. 

(a)  All  real  quantities  (positive,  negative,  and  zero)  may 
be  represented  by  the  points  of  a  directed  line  (running,  say, 
from  left  to  right) : 

0 0 0 0 0 0 0 > 

_3    __2  —1        0    +1    +2   +3 

and  the  notation  a<h  (read:  **a  algebraically  less  than  h") 
means  simply  that  a  precedes  &,  or  a  lies  on  the  left  of  h,  along 
this  line. 

Similarly,  o  >  6  (read:  *'a  algebraically  greater  than  h")  means  that 
a  comes  after  h,  or  lies  on  the  right  of  b,  along  the  line.  (The  idea  that 
a  negative  quantity  is  a  magnitude  whose  size  is  in  some  way  "less  than 
nothing''  should  be  carefully  avoided.) 

16 


ALGEBBA.  17 

Obviously,  if  a  and  h  are  any  real  quantities,  one  and  only 
one  of  the  three  relations :  a  =  h,  a<,b,  and  a>h,  will  hold 
between  them ;  further,  if  a  <  &  and  &  <  c,  then  a  <.c, 

(b)  Complex  quantities  require  for  their  representation  the  points  of 
a  plane  instead  of  the  points  of  a  line,  and  the  symbols  <  and  >  are  not 
used  in  connection  with  these  quantities. 

Legitimate  operations  on  inequalities.    If  a  given  inequality        j 
is  true,  it  will  still  be  true  if  we  / 

(a)  add  any  quantity  we  please  to  both  sides; 
(6)  subtract  any  quantity  we  please  from  both  sides; 

(c)  multiply  both  sides  by  any  positive  quantity; 

(d)  divide  both  sides  by  any  positive  quantity; 

(e)  raise  both  sides  to  any  positive  power  (integral  or 
fractional),  provided  both  sides  are  positive. 

(/)  take  the  logarithm  of  both  sides,  provided  both  sides 
are  positive. 

If  we  multiply  or  divide  both  sides  by  any  negative  number, 
we  must  reverse  the  sense  of  the  inequality. 

The  neglect  of  the  rules  for  handling  inequalities  is  the  source  of  many 
common  errors. 

19.  Arithmetical  Progression. 
In  an  arithmetical  progression : 

a,    a  +  d,    a  +  2d,    a  +  Sd,     •••, 

each  term  is  obtained  from  the  preceding  by  adding  a  con- 
stant quantity. 

The  nth  term  is  obviously  l  =  a-\-  {n  —  l)d. 

a  +  l 
The  sum  of  n  terms  is  S^  — ^p-  n. 

This  formula  is  most  easily  remembered  in  the  form: 

jSi=  (average  of  the  first  and  last  terms)  X  (number  of  terms). 
The  arithmetic  mean  between  a  and  b  isA  =  J(a-|-&). 

20.  Geometric  Progression. 
In  a  geometric  progression : 

a,    ar,    ar^y    ar',     •••, 


18  ALGEBRA. 

each  term  is  obtained  from  the  preceding  by  multiplying  by  a 
constant  quantity. 

The  nth  term  is  obviously  l  =  ar^-'^, 

\a{l  —  r«) 

The  sum  of  n  terms  is  ^=  — ^ . 

1  —  r 

This  formula   is  best   remembered  in   connection  with  the   rule  for 
factoring : 

1  —  r**  =  (1  —  r)  (1  +  r  +  r2  +  r»  H \-  r^-i). 

The  geometric  mean  between  a  and  h  is  G=y/ah. 
The  geometric  mean  is  also  called  the  mean  proportional. 

Infinite  geometric  progression.    If  |  r  |  <  1,  the  sum  of  n 
terms  approaches  the  limit 

a 


1  — r 


as  n  increases  indefinitely  (since,  in  the  expression  for  S,  if 
I  r  I  <  1,  r"  approaches  zero). 

21.  Harmonic  Progression. 

A  harmonic  progression  is  a  series  of  terms  whose  recip- 
rocals are  in  arithmetical  progression.  (The  harmonic  pro- 
gression is  not  of  great  importance.) 

The  harmonic  mean  between  a  and  h  is  H= -. 

a4-  6 


A  SYLLABUS  OF  ELEMENTARY  GEOMETRY  AND 
MENSURATION. 

This  syllabus  is  intended  to  include  those  facts  and  methods  of  ele- 
mentary geometry  which  a  student  should  have  so  firmly  fixed  in  his 
memory  that  he  will  never  think  of  looking  them  up  in  a  book. 

1.  Right  Triangles. 

In  a  right  triangle,  the  square  on  the  hypotenuse  is  equal 
to  the  sum  of  the  squares  on  the  other  two  sides  (Pythagoras, 
580-501  B.C.) ;  and  the  sum  of  the  acute  angles  is  90°. 

Examples  of  right  triangles  with  integral  sides:  3,  4,  5;  5,  12,  13. 

Two  right  triangles  are  congruent  when  they  agree  with 
respect  to  (a)  any  side  and  an  acute  angle;  or  (&)  any  two 
sides. 

In  the  **45°  triangle"  and  the  ''30-60°  triangle,"  the  ratios 
of  the  sides  are  as  indicated  in  the  figure. 


2.  Oblique  Triangles. 

In  any  plane  triangle,  the  sum  of  the  angles  is  180°.    Hence, 
an  exterior  angle  of  a  triangle  equals  the 
sum  of  the  opposite  interior  angles. 

Of  two  unequal  sides  in  a  triangle,  the  greater  is  opposite 
the  greater  angle. 

A  plane  triangle  is,  in  general,  wholly  determined  when  any 
three  of  its  parts   (not  all  angles)   are  given. 

19 


20  ELEMENTAEY    GEOMETBY    AND    MENSUBATION 

There  are  four  cases : 


'^11     ff^ 


(a)  two  angles    (provided  their  sum  is  less  than 
180")  and  one  side; 

(6)  two  sides  and  the  included  angle; 

(c)  the    three    sides     (provided    the 
largest  is  less  than  the  sum  of  the  other         -^^'''^fv.         ^^     {"^^ 
two) ;  "^^  "  ^^  "  1^^' 

(d)  two  sides  and  the  angle  opposite  one  of  them  (the  *' ambiguous 


Hence  the  usual  rules  for  testing  the  equality  of  two  plane 
triangles. 

The  center  of  gravity  of  a  plane  triangle  is  the 
intersection  of  the  three  medians,  and  is  two 
thirds  of  the  way  from  any  vertex  to  the  middle 
point  of  the  opposite  side. 

3.  Angles  in  a  Circle. 

An  angle  inscribed  in  a  semicircle  is  a  right  angle. 


^ 


An  angle  subtended  by  an  arc  of  a  circle  at  any  point  of  the 
circumference  is  equal  to  half  the  angle  subtended  by  the  same 
arc  at  the  center. 

4.  Similar  Figures.    Proportion. 

If  any  two  lines  are  cut  by  a  set  of  parallels, 
the  corresponding  segments  are  proportional. 
(Hence  the  usual  rule  for  dividing  a  given  line 
into  any  number  of  equal  parts.) 

In  all  problems  in  proportion,  the  notation  a:&::c:d,  and  all  speeial 
terminology,  such  as  ''taking  a  proportion  by  alternation,''  "by  com- 


ELEMENTABY    GEOMETRY    AND    MENSURATION. 


21 


positioii/*  etc.,  should  be  abandoned  in  favor  of  the  ordinary  language 
of  the  equation.  For  example,  if  a/6  =  c/d,  then,  by  adding  1  to  both 
sides,  (a  +  b)/h  =  {c  + d)/d;  and  by  subtracting  1  from  both  sides, 
(o  — 6)/&=(c  — d)/d;  etc. 

If  two  plane  triangles  are  similar,  their  corresponding  sides 
are  proportional. 

In  a  right  triangle,  the  perpendicular 
from  the  vertex  of  the  right  angle  to  the 
hypotenuse  is  a  mean  proportional  be- 
tween the  segments  of  the  hypotenuse: 


mn. 


Any  two  similar  fig- 
ures, in  the  plane  or  in 
space,  can  be  placed  in 
'*  perspective,*'  that  is, 
so  that  lines  joining 
corresponding  points  of  the  two  figures  will  pass  through  a 
common  point.  In  other  words,  of  two  similar  figures,  one  is 
merely  an  enlargement  of  the  other. 

In  two  similar  figures,  if  k  is  the  factor  of  proportionality, 
any  length  in  one  =  Aj  X  (the  corresponding  length  in  the 
other) ;  any  area  in  one  =  k^  X  (the  corresponding  area  in 
the  other) ;  any  volume  in  one  =  /c'  X  (the  corresponding  vol- 
ume  in  the  other) . 


5.  Lines  and  Planes. 

If  a  line  is  perpendicular  to  a  plane, 
every  plane  containing  that  line  is  perpen- 
dicular to  the  plane. 


7 


22 


BLEMBNTABY    GEOMETBY    AND    MENSURATION. 


A  dihedral  angle  is  measured  by  a  plane  angle 
formed  by  two  lines,  one  in  each  face,  perpen- 
dicular to  the  edge. 


6.  Plane  Areas. 
Area  of  parallelogram 

I  =  base  X  altitude. 

Area  of  triangle 

=  ^  base  X  altitude, 
^rea  of  trapezoid 
y^         ==  \  sum  of  II  sides  X  alt. 
==  mid-section  X  altitude. 

7.  The  Circle.     (7r.=  3.1416  •  •  •  =  22/7,  approximately.) 
Circumference  of  circle  =  27rr. 
(Proved  by  regarding  tbe  circle  as  the 

limit    of   an    inscribed    or   circumscribed 
polygon;  proof  rather  long.) 

Area  of  circle  =  irr^. 

(Proof  by  regarding  circle  as  limit  of  sum  of  triangles  radiating  out 
from  the  center,  the  altitude  of  each  triangle  being  the  radius  of  the 
circle;  hence,  area  of  circle  =i  circumference  X  radius.) 

Area  of  sector        angle  of  sector 


r\ 

m 

K 

A 

area  of  circle        four  right  angles  ' 


hence. 


Area  of  sector  =  \r'^B,  where  0  is  the  angle  in  radians. 
For  area  of  segment,  subtract  triangle  from  sector. 


f^ 


\ 


ELEMENTARY    GEOMETBY    AND    MENSURATION. 


23 


8.  The  Cylinder. 

Volume  of  any  cylinder  (or 
prism  )=  base  X  altitude. 

Area  of  curved  surface  of  any 
fight  cylinder  (or  right  prism)  = 
perimeter  of  base  X  altitude. 


(Proof  by  regarding  the  area  as  the  limit  of  a  sum  of  rectanglet 
whose  common  altitude  is  the  altitude  of  the  cylinder;  or,  by  slitting 
the  cylinder  along  an  ** element'*  and  rolling  the  surface  out  into  a 
rectangle.) 


t 


9.  The  Cone. 

Volume  of  any  cone  (or  pyra- 
mid) =  1/3  base  X  altitude. 

(Proof  by  dissecting  a  triangular 
prism;  or,  more  simply,  by  the  in- 
tegral calculus.) 


Area  of  curved  surface  of  a  right  circular  cone  (or  a  regular 
pyramid)  =  1/2  perimeter  of  base  X  slant  height. 

(Proof  by  regarding  the  area  as  the  limit  of  a  sum  of  triangles  whose 
common  altitude  is  the  slant  height  of  the  cone.) 

Area  of  frustum  of  a  right  circular  cone  (or  of  a  regular 
pyramid) 

=1/2  sum  of  perimeters  of  bases  X  slant  height. 
=  perimeter  of  mid-section  X  slant  height. 

(Proof  by  regarding  the  area  as  the  limit  of  the  trapezoids  whose 
common  altitude  is  the  slant  height  of  the  frustum.) 


24  ELEMENTAEY    GEOMETBY    AND    MENSURATION. 

10.  The  Sphere. 

Area  of  a  zone  =  circumference  of  great  circle  X  altitude 
of  zone. 

In  other  words,  the  area  of  the  sphere  cut  out  by  two  parallel  planes 
is  equal  to  the  area  of  the  portion  of  the  circumscribing  cylinder  inter- 
cepted between  the  same  pair  of  parallel  planes.  (Proof  by  regarding 
the  zone  as  the  limit  of  a  sum  of  conical  frustums.)     Hence^ 


!«: 


Area  of  sphere  =  47rr2 

=  area  of  four  great  circles  of  the  sphere. 

In  other  words,  the  area  of  the  sphere  is  equal  to  the  area  of  the 
rved  surface  of  the  circumscribing  cylinder. 


Volume  of  sphere  =  |Tr^. 

(Proof  by  regarding  sphere  as  limit  of  a  sum  of  pyramids  radi- 
ating out  from  the  center,  the  altitude  of  each  pyramid  being  the  radius 
of  the  sphere;  hence,  volume  of  sphere  =  J  area  of  sphere  X  radius.) 

Area  of  a  lune       angle  of  lune 


area   of  sphere       four  right  angles* 

Area  of  spherical  triangle  is  proportional  to    its 

spherical  excess  (that  is,  the  excess  of  the  sum  of  its 

angles  over  180°). 

(Proof  by  considering  three  lunes  which  have  the  given  triangle  in 
common.) 


ELEMBNTAEY    GEOMETBY    AND    MENSUEATION. 


25 


The  following  further  theorems,  the  proof  of  which  involves  the  inte- 
gral calculus,  are  mentioned  here  also,  because  they  are  easy  to  remember 
and  are  often  serviceable  in  elementary  work. 

11.  Cavalieri's  Theorem  (1598-1647). 

Suppose  two  solids  have  their  bases  in  the  same  plane,  and 
let  the  sections  made  in  each  solid  by  any  plane  parallel  to  the 
base  be  called  ** corresponding  sections.*'  If  then  the  corre- 
sponding sections  of  the  two  solids  are  always  equal,  the  vol- 
umes of  the  solids  will  be  equal. 

(Proof  by  regarding  each  of  the  solids  as  the  limit  of  a  pile  of  thin 
slabs.) 


12.  Theorems  of  Guldin   (1577-1643),  or 
of  Pappus  (about  290  A.D.). 

1.  Suppose  a  plane  figure  revolves  about 
an  axis  in  its  plane  but  not  cutting  it. 
Then  the  volume  of  the  solid  thus  generated 
is  equal  to  the  area  of  the  given  figure 
times  the  length  of  the  path  traced  by  its 
center  of  gravity. 

2.  Suppose  a  plane  curve  revolves 
about  an  axis  in  its  plane  but  not  cutting 
it.  Then  the  area  of  the  surface  thus 
generated  is  equal  to  the  length  of  the 
given  curve  times  the  length  of  the  path 
traced  by  its  center  of  gravity. 


26 


ELEMENTAKY   GEOMETRY  AND   MENSURATION. 


13.  The  Prismoidal  Formula. 

The  prismoidal  formula  holds  for  any  solid  lying  between  two  parallel 
planes  and  such  that  the  area  of  a  section  at  distance  x  from  the  base  la 
expressible  as  a  polynomial  of  the  second  (or  third)  degree  in  x. 

It  A,  B  =  areas  of  the  bases,  M  =  area  of  a  plane  section 
midway  between  the  bases,  and  h  =  altitude,  then 

Volume  of  prismoid  =  -  ( A  +  B  +  AM) . 


A  SYLLABUS  OF  PLANE    TRIGONOMETRY. 

This  syllabus  is  intended  to  include  those  facts  and  methods  of  plane 
trigonometry  which  a  student  should  have  so  firmly  fixed  iii  his  memory 
that  he  will  never  think  of  looking  them  up  in  a  book. 

Table  op  Contents. 
Chapter  I.    Sine,  Cosine,  and  Tangent  op  Acute  Angles. 

Definitions  of  sine,  cosine,  and  tangent  of  an  acute  angle  as  ratios 
between  the  sides  of  a  right  triangle: 

sin  a;  =  opp/hyp;  cos  a;:=adj/hyp;  tan  a;  =  opp/adj. 
To  trace  the  changes  in  these  functions,  as  the  angle  changes  from 
0"  to  90**   (circle  of  reference). 

Use  of  tables.    Exact  values  of  functions  of  30°,  45*,  and  60°. 
To  find  remaining  functions  of  an  angle  when  one  function  is  given 
(draw  right  triangle).    To  construct  an  angle  from  its  tangent. 
Fundamental  relations :  sin'  x  +  cos*  re  =  1,  tan  x  =  sin  x/coe  x,  etc. 
Solution  of  right  triangles. 
Problems  in  orthogonal  projection. 
Problems  in  composition  and  resolution  of  forces,  etc. 
Chapter  II.    The  Trigonometric  Functions  op  any  Angle. 

Angles  in  general.  Congruent,  complementary,  and  supplementary 
angles. 

Units  of  angular  measurement:  degree,  grade,  radian. 
Definitions  of  sine,  cosine,  and  tangent  of  any  angle. 
To  trace  the  changes  in  these  functions,  as  the  angle  changes  from 
0°  to  360°  (circle  of  reference). 
.  Definitions  of  cotangent,  secant,  and  cosecant: 

cot  a;  =  l/tan  x,    sec  ar  =  l/cos  x,    esc  a:  =  l/sin  x. 
Definitions  of  versed  sine  and  coversed  sine: 

vers  a?  =  1  —  cos  af;    covers  x  =  l  —  sin  x. 
Use  of  the  tables:  reduction  to  first  quadrant. 
Solution  of  oblique  triangles. 
Law  of  sines:  o/6  =  sin  -<4/sin  B. 
Law  of  cosines :    a'  =  6'  +  c*  —  2fec  cos  A. 
Chapter  III.    General  Properties  op  the  Trigonometric  Functions. 
Eelations  between  the  functions  of  a  single  angle. 
Functions  of  ( — x).    Functions  of  (a;±n90°),  etc. 
Functions  of  the  sum  and  difference  of  two  angles: 

sin  (a;  4"  y)  =  sin  a;  cos  y  +  cos  x  sin  y, 
cos  (a;  +  y)  =  cos  a;  cos  y  —  sin  x  sin  y. 
Functions  of  twice  an  angle,  and  of  half  an  angle. 
The  inverse  functions,  sin-^a;,  cos-^a;,  tan-*a:,  etc.       / 
Solution  of  trigonometric  equations. 

27 


CHAPTER  I. 


Sine,  Cosine,  and  Tangent  op  Acute  Angles. 

1.  Definition  of  sine,  cosine,  and  tangent  of  an  acute  angle 

X. — In  any  right  triangle,  if  x  is  one  of  the  acute  angles,  the 

sine,  cosine  and  tangent  of  x  are  defined  as  ratios  between  the 

sides  of  the  triangle,  as  follows: 

side  opp.  side  adj.  j"!^^ , 

sina;=— ; v^  cosa;=-^j ~  vi5>^ 

hypot.  y^ 

side  opp. 


hypot. 
tanaj= 


Fig.  2. 


side  adj.  Fia.  1. 

These  ratios  are  pure  numbers,  depending  only  on  the  size  of 
the  angle. 

2.  To  trace  the  changes  in  these  num- 
bers when  the  angle  changes  from  0**  to 
90°,  draw  the  figure  so  that  the  denomi- 
nator of  the  ratio  is  kept  constant,  say 
equal  to  1  inch,  and  trace  the  changes  in 
the  numerator.  Thus,  from  Fig.  2,  when 
X  goes  from  0°  to  90°,  sin  x  goes  from 
0  to  1,  and  cos  x  goes  from  1  to  0 ;  from 
Fig  3,  when  x  goes  from  0°  to  90°,  tan  x 
goes  from  0  to  infinity. 

3.  Tables. — The  ratios  thus  defined 
are  called  ** trigonometric  functions*' 
of  the  angle,  and  their  values  have 
been  tabulated,  to  4,  5,  or  6  places  of 
decimals,  in  the  **  tables  of  trigonometric 
functions.*'  Before  using  the  printed 
tables,  the  student  should  make  his  own 
table,  for  a  few  angles,  by  graphical  con- 
struction, with  a  protractor,  to  two 
places  of  decimals.* 

*  It  is  clear  from  the  figure  that  the  values  of  cos  x  from  0**  to  90°  are 
the  same  as  the  values  of  sin  x  in  reverse  order;  note  how  this  fact  is 
made  use  of  to  save  space  in  the  tables. 

28 


TRIGONOMETRY. 


29 


4.  The  functions  of  30°,  45°,  and  60°  can  be  found  exactly, 
without  the  use  of  the  table.  Thus,  in  the 
triangles  which  occur  in  Fig.  4,  it  is  readily 
proved  by  the  Pythagorean  theorem  that 
if  the  hypotenuse  is  1  inch,  the  shortest 
side  is  ^  in.,  the  longest  side  is  i^/S  in., 
and  the  middle-sized  side  JV2  in.  Hence 
any  function  of  30°,  45°,  or  60°  can  be  read 
off  the  figure  by  inspection.    For  example, 

sin  30°  =  i,     tan  45°  =:  1,    tan  60°  =  V3 ;     etc. 

5.  It  is  frequently  required  to  find  the  remaining  functions 


"l""-'-'-^ 

L 

/ 

M     1 

/ 

} 

<iff     ' 

\                    /  y 

V 

^ 

^ 

\! 

Fia. 

4. 

<i 


Fig.  5. 


of  an  angle  when  any  one  function  is  given.  To 
do  this,  draw  a  right  triangle,  mark  one  of  the 
angles,  and  mark  two  sides  to  correspond  to 
the  given  function.  Then  compute  the  remain- 
ing side  by  the  Pythagorean  theorem,  and  read 
off  any  desired  function  from  the  completed 
figure.     For  example. 

Given,  tan  x  =  ^.    From  the  figure,  sin  x  =  2/^/12;  etc. 

Given,  sin  x  =  a.    From  the  figure,  tan  x  =  a/y\  —  a^ ;  etc. 

To  construct  an  angle  when  any  one  of  its  functions  is 
given,  first  find  the  tangent  of  the  angle ;  when  the  tangent  is 
known,  the  construction  of  the  angle  is  obvious. 

6.  The  notation  sin^  x,  etc.,  is  used  as  an  abbreviation  for 
(sin  x)^',  etc. 

The  following  fundamental  relations  are  easily  proved  and 
remembered  from  the  figure :  for  any  angle  x, 


Fig.  7. 

sin^  X  +  cos^  x  = 
3 


1,    tan  X  = 


Fig.  8. 

sinic     sin  (90° — re)  =  cos  a;, 
cos  a;'    cos  (90° — ic)=sina;. 


30  TEIGONOMETRY. 

7.  The  student  should  be  thoroughly  drilled  in  the  defini- 
tions of  the  sine,  cosine  and  tangent,  in  right 
triangles  in  all  possible  positions  in  the  plane 
regardless  of  lettering.  Thus,  the  mental  proc- 
ess should  be  as  follows:  pointing  at  the  figure, 
''the  tangent  of  this  angle  is  this  side,  divided 
by  this  side'*;  etc. 

^'    '  The  following  forms  of  the  original  equations 

are  especially  useful,  and  should  be  emphasized: 

side  opp,  =  hypot.  X  sine ;    side  adj,  =  hypot.  X  cosine. 

Solution  op  Right  Triangles. 

8.  We  recall  that  in  any  right  triangle,  the  sum  of  the 
squares  on  the  two  legs  is  equal  to  the  square 
on  the  hypotenuse,  and  the  sum  of  the  acute 
angles  is  90°.     Hence,  when  either  acute  angle 

\  is  known,  the  other  may  be  found;  and  the 

sine  of  either  acute  angle  is  the  cosine  of  the 
other : 

c2  =  a2_|_52^     sinA  =  cos5. 

9.  By  the  aid  of  a  table  of  sines,  cosines  and  tangents,  when 
any  two  parts  of  a  right  triangle,  besides  the  right  angle,  are 
given,  the  remaining  parts  may  be  found  (except  in  the  case 
where  the  given  parts  are  the  two  acute  angles,  in  which  case 
the  triangle  is  not  determined). 

For,  we  have  merely  to  remember  the  definitions  of  the  func- 
tions, selecting  the  equations  so  that  only  one  unknown  ap- 
pears in  each  equation ;  then  solve  for  the  unknown  quantity, 
and  compute  by  the  aid  of  the  tables.  The  results  should  be 
checked  by  substituting  in  some  relation  not  used  in  the  direct 
computation.* 

*  This  computation,  like  many  other  numerical  computations,  can  often 
be  shortened  by  the  use  of  the  slide  rule,  or  by  the  use  of  logarithms; 
in  fact,  tables  are  provided  which  give  the  logarithms  of  the  trigono- 
metric functions  directly  in  terms  of  the  angles;  but  the  student  should 
thoroughly  understand  the  use  of  the  functions  themselves  before  he 
begins  to  use  the  logarithmic  tables. 


TRIGONOMETRY. 


31 


10.  Numerous  problems  involving  right  triangles :  isosceles 
triangles,  polygons,  oblique  triangles  solved  by  means  of  right 
triangles,  heights  and  distances,  surveying  problems,  etc. 


A*Acod'0 


Fig.  11. 


Orthogonal  Projection.     Components  of  Forces,  etc. 

11.  The  projection  of  a  length  AB  on  any  line  is  the  given 
length    times    the    cosine    of 
the  angle  between  the  lines. 
(Proof  from  the  definition  of 
cosine.) 

The  projection  of  a  plane 
area  upon  any  fixed  plane  is 
the  given  area  times  the  cosine 
of  the  angle  between  the 
planes.  (Proof  by  the  theo- 
rem of  limits.) 

'     12.  The  component  of  a  force  along  any  fixed  axis  is  the 
magnitude  of  the  force  times  the  cosine  of  the   y.  r^ 

angle  between  the  force  and  the  axis.  rv"'^^'^< 

Since   we   usually   require   the   components         pi^.  12. 
along  two  rectangular  axes,  it  is  important  to     y^  %_ 

remember   that   cos    (90°  —  rc)==sinic.      The       i*^--^ 
mental  process  should  be  as  follows :  Fig.  13. 

In  Fig.  12,  the  component  of  F  along  the  2/-axis  is  F  times 
the  cosine  of  0;  the  component  of  F  along  the  rc-axis  is  F 
times  the  cosine  of  the  other  angle,  which  is  F  times  the  sine 
of  $;  that  is,  Fy  =  FQOs0;  Faf  =  F  sin  6.  Similarly,  in  Fig. 
13,  Fx  =  F  COS  <f>;  Fy=^  —  Fsm<f>  (minus,  because  it  pulls 
backward  along  that  line) . 

The  components  of  velocities,  accelerations,  or  any  other 
vector  quantities  are  to  be  handled  in  the  same  way. 

13.  Every  problem  should  be  accompanied  by  a  sketch  or 
diagram,  to-shewiiiat  the  student  understands  the  meaning 
of  each  step  of  his  work.  And  in  many  cases,  an  accurate 
graphical  solution  on  a  drawing  board  may  be  used  as  a  valu- 
able check  on  the  correctness  of  the  numerical  computation. 


^ 


32  TRIGONOMETRY. 

14.  Note.  That  portion  of  trigonometry  which  has  been 
outlined  up  to  this  point  is  so  elementary  in  character,  and  so 
readily  understood  and  appreciated  by  the  student,  that  it 
may  well  he  introduced  much  earlier  in  the  course  than  is 
usually  done — perhaps  even  as  early  as  the  elementary  course 
in  plane  geometry. 


CHAPTER  II. 

The  Trigonometric  Functions  of  Any  Angle. 

15.  Angles  in  general. — An  angle,  as  the  term  is  used  in  ap- 
plied mathematics,  is  the  amount  of  rotation  of  a  moving 
radius  OP  about  a  fixed  point  0,  measured  from  a  fixed  line 


OX,  Here  OX  is  called  the  initial  line  and  OP  the  terminal 
line  of  the  angle.  Counterclockwise  rotation  is  positive,  and 
angles  are  added  and  subtracted  as  algebraic  quantities.  The 
quadrants  are  numbered  as  in  the  figure;  an  '* angle  in  quad- 
rant 77"  for  example,  means  an  angle  whose  terminal  line  lies 
in  quadrant  77. 

16.  Congruent  angles  are  angles  differing  by  any  multiple 
of  360°. 

17.  Complementary  angles  are  angles  whose  sum  is  90° ; 
supplementary  angles  are  angles  whose  sum  is  180°. 

18.  Units  of  angular  measurement  are:  the  degree,  sub 
divided  into  minutes  and  seconds,  or  decimally;  the  grade, 

33 


34  TRIGONOMETRY. 

subdivided  decimally;  and  the  radian,  subdivided  decimally. 

1  degree  =  V=  l/90th  of  a 
right  angle; 

1  grade  =  1/lOOth  of  a  right 
angle  (used  in  France) ; 

1  radian  =  angle  subtended  by 

an  arc  equal  to  the  radius.  Fig.  15. 

Since  ratio  of  semi-circumference  to  radius  =  7r  (where 
7r  =  3.1416  ...=3%  approximately),  we  have 

TT  radians  =  180°,  and  hence  1  radian  =  about  57.3°. 

19.  The  radian  is  especially  important  in  problems  concern- 
ing the  motion  of  a  particle  in  a  circular  path.     Thus,  if 

r  ft.  =  radius  of  the  circle, 

s  ft.  =  length  of  arc  traversed,  and 

$  radians  =  angle  swept  over  by  the  moving  radius,  then 

s  =  re. 

This  important  equation  is  not  true  unless  the  angle  is  meas- 
ured in  radians.     Again,  if 
V  ft.  per  sec.  =  linear  velocity  of  the  particle  in  its  path,  and 
(i>  radians  per  sec.  =  its  angular  velocity,  then 

V  =  ra>. 

Further,  if  the  angular  velocity  =  w  radians  per  sec.  =  N 
rev.  per  min.,  then  the  relation  between  the  numbers  w  and 
N  is  given  by 

ttN 

"=30- 

In  all  higher  mathematics,  when  a  letter  is  used  for  an 
angle,  without  designating  the  unit,  it  is  understood  that  the 
letter  means  the  number  of  radians  in  the  angle. 


TRIGONOMETRY. 


35 


20.  Definition  of  sine,  cosine,  and  tangent  of  any  angle. — 
Let  X  be  any  angle,  swept  over  by  a  moving  radius  revolving 
from  OX  to  OL,  and  suppose  for  convenience  of  language  that 
OX  extends  horizontally  to  the  right.  Assume,  for  the  moment 
that  OX  and  OL  are  not  perpendicular.  From  any  point  P 
of  the  moving  radius  drop  a  perpendicular  on  the  initial 
line  (or  the  initial  line  produced),  thus  forming  a  right  tri- 


FiG.  16. 

angle,  called  the  triangle  of  reference  for  the  given  angle  x. 
In  this  triangle,  the  perpendicular  MP  is  called  the  side  oppo- 
site 0,  and  is  positive  if  it  runs  up,  negative  if  it  runs  down ; 
the  base  OM  is  called  the  side  adjacent  to  0,  and  is  positive  if 
it  runs  to  the  right,  negative  if  it  runs  to  the  left,  and  the 
radius  OP  is  called  the  hypotenuse  of  the  triangle  and  may 
always  be  taken  as  positive.  The  sine,  cosine  and  tangent  of 
the  angle  x  are  then  defined  as  follows : 

side  opp.  side  adj.     ,  side  opp.       sin  x 

=  ^r T^,    cos  X  =  ^r —y    tan  X  =  -r^ — ~7-  = 

hypot.  hypot.  side  adj.       cos  a; 


sina;  = 


These  ratios  are  positive  or  negative  numbers,  depending 
only  on  the  position  of  the  terminal  side  of  the  angle  x,  and 


36  TRIGONOMETRY. 

are  called  trigonometric  functions  of  x.  The  functions  of  any 
angle  congruent  to  x  are  the  same  as  the  functions  of  x,  so 
that  we  need  consider  only  the  angles  in  *'the  first  revolu- 
tion,'* that  is,  angles  between  0°  and  360°. 

21.  To  trace  the  changes  in  each  function  as  the  angle 
changes  from  0°  to  360°,  draw  the  figure  so  that  the  denomi- 
nator of  the  ratio  is  kept  constant,  say  equal  to  1  inch,  and 
trace  the  changes  in  the  numerator  (Fig.  17  for  the  sine  and 
cosine;  Fig.  18  for  the  tangent).  Obviously,  the  sine  will  be 
positive  for  angles  in  the  upper  quadrants;  the  cosine  will 
be  positive  for  angles  in  the  right  hand  quadrants;  and  the 
tangent  will  be  positive  in  quadrants  I  and  III, 

The  definitions  of  the  functions  of  0°,  90°,  180°,  and  270°, 
which  were  not  included  above,  can  now  be  readily  obtained 
by  noting  what  becomes  of  the  function  of  a  variable  angle 
X  when  x  approaches  one  of  these  values  as  a  limit. 

In  using  the  ** circle  of  reference'*  be  careful  to  have  every 
angle  start  from  the  initial  line  that  extends  horizontally  to 
the  right. 

Other  Trigonometric  Functions. 

22.  Definition  of  other  trigonometric  functions. — ^Besides 
the  sine,  cosine,  and  tangent,  other  functions  in  common  use 
are  the  cotangent,  the  secant,  and  the  cosecant,  which  are 
most  conveniently  defined  thus : 

cot  X  =  7 ,     sec  X  =  ,     esc  x  = 


tan  X  cos  x^  sin  x 

Less  important,  but  often  convenient,  are  the  versed  sine  and 
the  coversed  sine: 

vers  x  =  l  —  cos  ic,    covers  x  =  l  —  sin  re. 

23.  It  is  worth  remembering  that  the  sine  and  cosine  are 
always  less  than  (or  eqtlal  to)  1,  in  absolute  value;  their 
reciprocals,  the  secant  and  cosecant,  are  always  greater  than 
(or  equal  to)  1,  in  absolute  value;  the  tangent  and  cotangent 
may  have  any  value,  positive  or  negative;  while  the  versed 
sine  and  coversed  sine  are  always  positive,  ranging  from  0  to  2. 


TRIGONOMETKY. 


37 


Fig.  17. 


Fig.  18. 


38 


TRIGONOMETRY. 


Draw  any 
(avoiding, 


X'^QT 


Fig.  19. 


24.  Use  of  the  tables:  reduction  to  the  first  quadrant. — The 
tables  in  common  use  give  the  values  of  the  functions  only 
for  angles  between  0°  and  90°,  that  is,  only  for  angles  in  the 
first  quadrant.  To  find  the  functions  of  an  angle  x  in  one 
of  the  other  quadrants,  find  first  the  ''reduced  angle"  in 
quadrant  I  (that  is,  x  — 90°,  or  a;  — 180°,  or  x  — 270°),  and 
then  proceed  as  in  the  following  examples:* 

(a)  To  find  cos  a;,  when  x  is  in  quadrant  II. 
angle  in  quadrant  II  to  represent  the  angle  x 
however,  lines  near  the  middle  of  the 
quadrant)  and  draw  the  "  reduced 
angle"  x  —  90°  in  quadrant  I.  Then, 
pointing  at  the  figure,  cos  x  is  this  line 
CvW")  [divided  by  the  radius],  which 
is  the  same  in  length  as  this  line  (<) 
[divided  by  the  radius],  which  is  the 
sine  of  x  —  90° ;  but  the  first  line  is 
negative;  hence 

cos  ic  :^  —  sin  {x  —  90** ) , 

where  sin  {x  —  90°),  of  course,  can  be  found  in  the  table. 

(6)  To  find  tana;,  when  x  is  in  quadrant  II.  Pointing  at 
the  figure,  tana;  is  this  line  (<)  divided 
by  this  line  (|||),  which  is  the  same  as 
this  line  (VVV)  divided  by  this  line  (^), 
which  is  the  cotangent  of  {x  —  90°); 
but  the  signs  are  unlike;  hence 

tan  a;  =  — cot  (a;  — 90°), 

where  cot  {x  —  90°)  can  be  found  from 
the  table. 

Similarly  for  any  other  case. 

25.  The  converse  problem  of  finding  the  angle  correspond- 
ing to  any  given  function  is  complicated  by  the  fact  that  there 
will  be  (in  general)  two  angles  between  0°  and  360°  corre- 
sponding to  any  given  function.    The  most  satisfactory  way 

*  The  given  angle  is  supposed  to  be  already  reduced  to  an  angle  be- 
tween O**  and  360°. 


Fig.  20. 


TRIGONOMETRY. 


39 


to  find  these  two  angles,  in  any  numerical  case,  is  to  draw 
the  figure,  and  proceed  as  in  the  examples  below,  in  which  Xq 
in  each  case  represents  an  angle  in  the  first  quadrant  which 
can  be  found  in  the  table. 


Fig.  21. 


Given  sin  a;  =  0.5; 
x  =  Xq  or  180°  —  Xr 


Fig.  23. 


Given  cos  a;  =  0.8; 
x  =  Xr,  or  360°— :3 


Fig.  25. 

Given  tan  a;  =  0.8; 
x  =  Xq  or  180°+a;o. 


Given  sina;= — 0.5; 

a;  =  180°+a;o  or  360°— Xq. 


■CO>^--0-5 


Fig.  24. 
Given  cosa;  =  —  0.5; 
a;  =  180°— rco  or  180°+a;o. 


Given  tanic  =  —  0.8; 

a;  =  180°— a^o  or  360°— a^o- 


40  TRIGONOMETRY. 

These  results  are  not  formulae  to  be  memorized;  it  is  much 
safer,  and  more  intelligent,  to  draw  the  appropriate  figure, 
or  to  visualize  it  in  the  mind,  for  each  case  as  it  arises.  The 
student  should  be  thoroughly  drilled  in  numerical  cases, 
especially  for  angles  in  the  second  quadrant. 

Notice  that  an  angle  is  completely  determined  when  we 
know  the  value  of  any  one  of  its  functions,  and  the  sign  of 
any  other  function  (not  the  reciprocal  of  the  first). 

It  we  restrict  ourselves  to  angles  between  0°  and  180°, 
as  in  the  case  of  angles  in  a  triangle,  then  an  angle  is  wholly 
determined  by  either  its  cosine  or  its  tangent;  but  there  will 
be  two  angles,  x  and  180°  — x,  corresponding  to  a  given  sine. 

26.  The  functions  of  certain  angles  in  the  later  quadrants, 
corresponding  to  30°,  45°,  and  60°  in  quadrant  I,  may  be 
found  exactly,  without  the  use  of  the  tables,  by  inspection 
of  the  figure  (see  §  4). 

For  example,  cos  120°=  — J. 

27.  If  it  is  required  to  find  the  remaining  functions  of  an 
angle  when  one  function  is  given,  draw  a  right  triangle  and 
proceed  as  in  §  5,  considering  only  the  absolute  values  of 
the  quantities,  without  regard  to  sign;  then  adjust  the  sign 
of  the  answer  according  to  the  quadrant  in  which  the 
angle  lies.  Or,  the  angle  may  be  drawn  at  once  in  the  proper 
quadrant. 

Solution  of  Oblique  Triangles. 

28.  In  any  plane  triangle  the  following  theorems  are  easily 
proved  from  a  figure : 

(1)  The  ^^Law  of  Sines  J' — Any  side  is  to  any  other  side  as 
the  sine  of  the  angle  opposite  the  first  side  is  to  the  sine  of  the 
angle  opposite  the  other  side ;  in  the  usual  notation : 

a      sin  A 


b      sin  B' 

with  two  analogous  formulae  obtained  by  "advancing  the 
letters." 


TRIGONOMETRY.  41 

(2)  The  ^^Law  of  Cosines.' ' — The  square  of  any  side  is 
equal  to  the  sum  of  the  squares  of  the  other  two  sides,  minus 
twice  their  product  times  the  cosine  of  the  included  angle : 

a2  =  62  _(_  c2  —  2hc  cos  A, 

with  two  analogous  formulae  obtained  by  **  advancing  the 
letters." 

These  two  laws,  with  the  fact  that  the  sum  of  the  angles  is 
180°,  suffice  to  ** solve"  any  plane  triangle,  and  are  important 
in  many  theoretical  considerations. 

The  following  formulae  which  are  especially  adapted  to 
logarithmic  computation,  give  the  tangents  of  the  half -angles 
in  terms  of  the  sides,  and  are  included  here  for  reference : 

^      A  r  ^      B  r  ^      C         r 

tan  -  = ,         tan  -  = j ,         tan  -  = 

2       s  — a'  2       s  —  6  2       s  —  c 

where 


a  -\-  b  -\-  c 


and 


-4 


i? — ^zS^ ISl — y  =  radius  of  inscribed  circle. 

From  these  formulae  we  have  at  once, 


Area  =  rs=  V^C^  —  ^)  (^  —  ^)  (^  —  c). 

29.  The  only  case  which  is  likely  to  give  any  difficulty,  is 
the  ** ambiguous  case"  in  which  the  given  parts  are  two  sides 
and  the  angle  opposite  one  of  them.  Here  we  must  remem- 
ber, at  a  certain  point  in  the  work,  that  when  the  sine  of  an 
angle  is  given,  there  will  be,  in  general,  two  angles  corre- 
sponding to  that  sine,  one  the  supplement  of  the  other;  so 
that  from  that  point  on,  the  problem  breaks  up  into  two 
separate  problems.  But  if  the  sine  of  an  angle  is  1,  then 
the  only  value  for  the  angle  is  90° ;  and  if  the  sine  is  greater 
than  1,  there  is  no  corresponding  angle,  and  the  problem  is 
impossible.  It  is  advisable  to  construct  a  fairly  accurate 
figure. 


42  TKIGONOMETKY. 

30.  Problems  in  oblique  triangles,  triangulation,  etc. 

In  every  case  at  least  a  rough  sketch  should  be  drawn  on 
which  the  known  parts  are  clearly  marked,  and  a  **  blank 
form*'  for  the  computation  should  be  made  out  for  the  entire 
problem,  before  any  of  the  quantities  are  looked  up  in  the 
table. 


CHAPTER   III. 
General  Properties  of  the  Trigonometric  Functions. 

31.  Relations  between  the  functions  of  a  single  angle, — The 
student  should  convince  himself  that  the  following  important 
relations  will  hold  for  any  angle  x: 


sin'  X  -j-  cos'^  x=l,     tan  x  = 
sec'  x=l  -\-  tan'  x. 


sin  a; 
cosa;' 


All  these  relations  are  easily  recalled  by  the  aid  of  the 
figures. 

Somewhat  less  important  is  the  following : 
csc^x  =  1  +  QOt^X, 

32.  Functions  of  ( — x).    From  the  figure, 
sin  ( — x)  =  —  sinx, 
cos  ( — ic)=cosic, 
tan  ( — x)  =  —  tana;. 

33.  Functions  of  (90°  +  x) ,  (x +  180°),  etc.— Any  iunQiion 

of  a   combination  like    {xztn90°)    or    (n90°  ±:x)    can  be 

expressed  in  terms  of  a  function  of  x  by  the  use  of  the  figure. 

For  example,  find  sec  (270°  —  x).     Take  as  x  any  small 

angle  in  the  first  quadrant,  and  draw  the  angle  270° — x. 

Then,    sec    (270° —  a;)    is    1    over   the 

cosine  of  (270°  — x),  which,  pointing  at 

the  figure,  is  the  radius  over  this  line 

(  VVV  ),  which  is  the  same,  in  length,  as 

the  radius  over  this  line  ( ^  ) ,  which  is 

1  over  the  sine  of  x,  or  esc  x.    But  the 

signs  are  opposite ;  therefore, 

sec  (270°— a;)  =— CSC  a;. 
43 


2lO^X 


44  TRIGONOMETEY. 

This  method  requires  the  memorizing  of  no  rules  or  for- 
mulae, besides  the  definitions  of  the  functions;  a  very  little 
practice  will  develop  all  the  speed  and  accuracy  that  can  be 
desired,  and  the  method  is  one  which  is  readily  recalled  to 
mind  after  long  disuse.  The  special  case  of  complementary 
angles,  however,  is  worth  remembering  as  a  separate  formula : 
Any  function  of   (90° — ic)  =the  co-named  function  of  x. 

Formulas  for  the  Sum  op  Two  Angles,  Etc. 

34.  In  simplifying  trigonometric  expressions  which  occur 
in  calculus,  mechanics,  etc.,  the  following  formulae  are  so  fre- 
quently required  that  they  should  be  thoroughly  memorized. 
The  ability  to  recognize  those  relations  readily,  regardless  of 
the  special  lettering  employed,  is  a  necessary  condition  for 
rapid  progress  in  almost  any  branch  of  analysis,  but  it  is 
highly  undesirable  to  extend  the  list  beyond  the  limits  here 
given. 

The  fundamental  formulae  from  which  all  others  are  derived 
are  these  two,  the  proof  of  which  is  obtained  from  a  figure : 

(1)  sin  {x  -{-y)  =  sin  xcosy  -{-  cos  x  sin  y, 

(2)  cos  (x  +  2/)  =  cos  a;  cos  2/  —  sin  x  sin  t/. 

These  and  the  following  formulae  should  be  memorized  in 
words,  not  in  letters :  thus,  *  *  the  sine  of  the  sum  of  two  angles 
is  the  sine  of  the  first  times  the  cosine  of  the  second,  plus  the 
cosine  of  the  first  times  the  sine  of  the  second, '  *  etc. 

Dividing  (1)  by  (2)  and  then  dividing  numerator  and  de- 
nomerator  by  the  product  of  the  cosines,  we  have 

(3)  tan  (x  -\-  y)  = :r—' 

^   ^  V     '  :/^       1  —tana;  tan  ^ 

Changing  the  sign  of  y  in  these  three  formulae,  and  remem- 
bering the  relations  for  negative  angles,  we  have  the  corre- 
sponding formulae  for  sin  (x  —  2/)»  cos  (ic  —  y),  tan  (a;  —  i/), 
which  will  be  exactly  the  same  as  (1),  (2),  and  (3)  with  all 
the  connecting  signs  reversed: 


TBIGONOMETBY.  45 

(4)  sin  {x  —  t/)  =:  sin  a;  cos  y  —  cos  re  sin  y, 

(5)  cos  (x  —  y)  =  cos  aj  cos  2/  +  sin  a;  sin  y, 

(6)  tan  (ic  —  y)  =  zr-T-:: r— ^. 

^   ^  ^         ^^       1  +  tan  ar  tan  y 

Putting  x  =  y  in  (1),  (2),  and  (3)  we  have  at  once 

(7)  sin  2a;  =  2  sin  x  cos  rr, 

(8)  cos  2x  =  cos^  a;  —  sin^  x 

=  1  —  2  sin'  a;  =  2  cos'  a;  —  1, 

2  tan  X 


(9)  tan  2x 


1  —  tan*  x' 


Solving  (8)  first  for  sin  a?  and  then  for  cos  a;,  and  putting 
2a;  =  2/,  or  x  =  y/2j  we  find 


(10) 

-1=^/-^=!^''* 

(11) 

whence, 

C0S|=:.^1±|^. 

x-irtN  X      y       _i_     /I  —  cos  y 

(12)  tan^=±^-_ ^. 

^  2  \  1  +  cos  y 

This  last  formula  may  be  transformed,  by  rationalizing 
numerator  or  denominator,  into 

y       1  —  cos  y  sin  y 

tan  ^  = ^ ^  =  -— — ^— . 

2  sm  2/  1  4-  cos  y 

Other  formulas,  useful  for  special  purposes,  should  not  be 
memorized,  but  should  be  derived  as  needed. 

35.  In  proving  the  identity  of  two  trigonometric  expres- 
sions, it  is  best  to  reduce  each  expression  separately  to  its 
simplest  form. 

*  The  plus  sign  is  to  be  used  when  sin  ^y  is  positive,  the  minus  sign 
when  sin  iy  is  negative.     Similarly  in  the  next  two  formulas. 
4 


46  TRIGONOMETRY. 

The  fallacy  of  supposing  that  because  a  true  relation  can  be 
deduced  from  a  given  equation,  the  given  equation  is  there- 
fore necessarily  true,  should  be  carefully  explained. 

For  example,  from  the  false  equation  3  =  —  3  we  can  obtain  the  true 
equation  9  =  9  by  squaring  both  sides;  or,  from  the  false  equation 
30**  =  150"  we  can  obtain  the  true  equation  %  =  ^^  by  taking  the  sine 
of  both  sides;  but  in  each  of  these  cases  the  step  taken  is  not  reversible. 

36.  The  following  device  for  transforming  an  expression 
of  the  form  a  cos  x  +  &  sin  x  is  often  useful : 

—  ^COS  (re  —  B)j 
where  A  =  \/{a^  +  6*)  and  tan  ^  =  -. 

Oi 

37.  The  inverse  functions. 

The  angle  between  — 90°  and  +90°  whose  sine  is  x  is  de- 
noted by  sin"^  x* 

The  angle  between  0°  and  180°  whose  cosine  is  a;  is  denoted 
by  cos~^  X. 

The  angle  between  —  90°  and  +90°  whose  tangent  is  x 
is  denoted  by  tan-^  x. 

In  simplifying  expressions  involving  these  **  inverse  func- 
tions,*' it  is  well  to  take  a  single  letter  to  stand  for  each  in- 
verse function ;  as,  2/  =  sin-^  x,  whence,  by  definition,  sin  y=x ; 
etc. 

38.  Solution  of  trigonometric  equations.  Many  trigonomet- 
ric equations  can  be  solved  only  by  the  ''method  of  trial  and 
error."  In  other  cases,  however,  it  is  possible,  by  the  use  of 
the  formulas  given  above,  to  transform  the  given  equation  into 
a  form  involving  only  a  single  function  of  a  single  angle; 
if  this  equation  can  be  solved  for  the  function  in  question, 
then  the  required  value  (or  values)  of  the  angle  can  be  found 
from  the  tables  or  it  can  be  shown  that  no  solution  exists. 

*  The  symbol  sin-*  a;  (or  arc  sin  x)  is  often  defined  as  simply  **the 
angle  whose  sine  is  a; '  * ;  but  since  there  are  many  such  angles,  it  is  neces- 
sary to  specify  which  one  is  to  be  taken  as  'Hhe"  angle,  if  the  symbol 
is  to  have  any  defijiite  meaning. 


A  SYLLABUS  OF  ANALYTIC  GEOMETRY. 

This  syllabus  is  intended  to  include  those  facts  and  methods  of  ana- 
lytic geometry  which  a  student  who  has  completed  an  elementary  course 
in  that  subject  should  have  so  firmly  fixed  in  his  memory  that  he  will 
never  think  of  looking  them  up  in  a  book. 

A  course  of  study  in  analytic  geometry  should  consist  chiefly  of 
problems  solved  by  the  students,  each  problem  being  solved  on  the  basis 
of  a  small  number  of  fundamental  formulas  such  as  are  here  mentioned. 

This  syllabus  is  confined  mainly  to  the  conic  sections;  but  a  satis- 
factory course  in  analytic  geometry  should  include  also  the  study  of 
many  other  curves,  both  in  rectangular  and  in  polar  coordinates.  The 
syllabus  takes  up  only  those  properties  of  curves  which  can  be  readily 
investigated  without  the  aid  of  the  calculus;  but  the  present  tendency 
to  introduce  the  elements  of  the  calculus  before  any  elaborate  study 
of  geometry  is  attempted  is  to  be  much  encouraged. 

Table  op  Contents. 

Chapter  I.    Eectangular  Co-ordinates. 

Chapter  II.    The  Straight  Line:  Equations  of  the  Form 

Ax-\-By  +  C  =  0. 
Chapter  III.    The  Circle:  Equations  of  the  Form 

x'-\-y^  +  Dx-\-Ey  +  F  =  0. 
Chapter  IV.     The  Parabola:  y'^=z2px. 
Chapter  V.     The  Ellipse  :   fe^^  +  a'y^  =  a^b^. 
Chapter  VI.     The  Hyperbola:  b^x^  —  a^'jf=zo?b^. 
Chapter  VII.    Transformation  of  Co-ordinates. 
Chapter  VIII.    General  Equation  of  the  Second  Degree  in 

X  and  y. 
Chapter  IX.    Systems  of  Conics. 
Chapter  X.    Polar  Co-ordinates. 
Chapter  XI.    Co-ordinates  in  Space. 


47 


CHAPTER  I. 

RECTANGULAR  COORDINATES. 


1,  In  many  geometrical  problems  it  is  convenient  to  describe 
the  position  of  a  point  in  a  plane  by  giving  its  distances  from  two 
fixed  (perpendicular)  lines  in  the  plane.* 

For  example,  on  a  map,  the  distance  of  a  point  to  the  east  or 
west  from  a  fixed  meridian  is  called  the  longitude  of  the  point,  and 
its  distance  north  or  south  from  the  equator  is  called  its  latitude. 

So  in  general,  in  any  plane,  the  distance  of  a  point  to  the  right 
or  left  from  a  fixed  vertical  axis  is  called  the  abscissa,  x,  of  the  point, 
and  its  distance  up  or  down  from  a  fixed  horizontal  axis  is  called 
its  ordinate,  y.  The  x  and  y  together  are  called  the  coordinates  of 
the  point. 

^  The  value  of  x  (=  OM)  will  be  positive  to  the 
right,  negative  to  the  left ;  the  value  of  y  {=MP) 
will  be  positive  upward,  negative  downward.  The 
point  for  which  x  =z  Xi  and  y  =i  y^  is  denoted  by 
Pi,  or(xi,  i/i). 


if 


0 


M 


2.     To  express  the  distance  between  two  'points  in  terms  of  their 
coordinates :         .v         , .  ^  ^  > . 


D=yJ{x,  —  x{)'  +  {y2  —  yi)\ 


3.     To  find  the  coordinates  of  the  point  half  way  between  two 


given  points:  ^fx^i  v  uam-  'Ju/yr^^  i 

y 

X  =  i(Xi  +  X2), 

y  =  K2/1  +  2/2). 

p 

^ 

^ 

a> 

*  We  restrict  ourselves  here  to  rectangular  axes;  oblique  axes  are,  however, 
occasionally  useful. 

48 


ANALYTIC  GEOMETRY. 


49 


4.  To  find  the  coordinates  of  a  point  P  on  the  line  through  two 
fixed  points,  and  such  that  its  distance  from  the  first  point  is  n 
times  the  distance  between   the  two    points     y 

(P?  =  nP^2): 

X  =  a^i  +  n{x2  —  Xi), 

y=yi+  n(2/2  —  2/i). 
Here  n  may  be  any  real  number  (positive,  negative,  or  zero). 

5.  To  find  the  slope  of  a  line  through  two  given  points: 


m  =  tan  <^  =  ^ 


2/1 


Xz  —  Xi 

The  angle  (j>  is  called  the  inclination 
of  the  line;  tan  </>  is  the  slope. 


6,     If  two  lines  are  parallel,  their  slopes  are  equal :  mi=zm2. 

V 


If  two  lines  are  perpendicular, 
the  product  of  their  slopes  is  minus  one: 
iM^Wg  =  — 1 . 


7.  To  express  the  areas  of  triangles  and  polygons  in  terms  of  the 
coordinates  of  the  vertices,  consider  the  trapezoids  formed  by  the 
ordinates  drawn  to  the  vertices. 

8.  In  any  problem  involving  an  unknown  point,  remember  that 
two  conditions  are  necessary  to  determine  the  coordinates  of  the  point 
(simultaneous  equations  in  two  unknown  quantities). 


CHAPTER  II. 

THE   STRAIGHT   LINE:  EQUATIONS    OF   THE   FORM  Ax  -{•  By -{-  C  =  0, 

9.  We  have  seen  that  if  two  conditions  are  imposed  on  x  and  y, 
the  position  of  the  point  (x,  y)  is  wholly  determined.  If  only  one 
condition  is  imposed,  the  position  of  the  point  is  only  partially 
restricted.     (Examples:  a:  =  5,  x^  +  !/^  =  25,  etc.) 

The  collection  of  all  points  which  satisfy  a  given  condition  im- 
posed on  X  and  y  is  called  the  locus  of  that  condition;  and  the  con- 
dition itself,  expressed  in  algebraic  language,  is  called  the  equation 
of  the  locu^.  Thus,  the  equation  of  a  straight  line  is  the  algebraic 
expression  of  the  condition  which  x  and  y  must  satisfy  in  order  that 
the  point  (x,  y)  shall  lie  on  the  line;  in  other  words,  the  equation  of 
a  line  is  an  equation  which  is  true  when  the  coordinates  of  any  point 
on  the  line  are  substituted  for  x  and  y,  and  false  when  the  coordinates 
of  any  point  off  the  line  are  substituted  for  x  and  y;  and  so  in  general 
for  the  equation  of  any  locus. 

10.  To  find  the  coordinates  of  the  points  of  intersection  of 
two  loci  whose  equations  are  given,  we  have  simply  to  find  the 
pairs  of  values  of  x  and  y  (if  any)  which  satisfy  both  the  equations 
at  once  (simultaneous  equations  in  x  and  y), 

11.  To  find  the  equation  of  a  line  (not  perpendicular  to  either 
axis),  when  its  slope,  m,  and  the  coordinates  of 
one  of  its  points  {xi,  2/1),  are  given: 

y  —  yi=  m{x  —  Xi). 

The  equation  of  a  line  perpendicular  to 
the  X-axis  (or  the  y-axis)  is,  by  inspection, 
x=  a{oT  y  =  b). 

The  equation  of  any  straight  line  is  of  the  form  Ax  +  By-}-  C  =  0,  and 
the  locus  of  every  equation  of  the  form  Ax -{-  By  -\-  C  =  0  is  a  straight  line. 
Hence,  to  plot  the  locus  of  such  an  equation,  it  is  sufficient  to  find  any  two  of 
its  points. 

12.  To  find  the  slope  of  a  line  whose  equation  is  given  (the  line 
being  not  perpendicular  to  an  axis),  write  the  equation  in  the 
form  y  =  {  )  ^+  {  )',  then  the  coefficient  of  x  will  be  the  slope. 

60 


ANALYTIC  GEOMETRY. 


51 


13.  To  find  the  equation  of  a  line  parallel  or  perpendicular  to  a 
given  line  and  through  a  given  point,  remember  that  rrii  =  nh  for 
parallellines,  and  miTrh  =  —  1  for  perpendicular  lines  (see  §  6). 

Special  method:  if  the  given  line  is  Ax  +  By-\-C  =  0,  then  the  parallel 
ia  Ax -\-  By  =  k  and  the  perpendicular  is  Bx  —  Ay  =i  K,  where  k  and  K  are  to 
be  determined. 

14.  To  find  the  angle   6   between  two  lines 
whose  slopes  are  given: 

tan  0  =  :i— 1 .* 

1  +  ?n2  Wi 

15.  To  find  the  distance  between  a  given  point  (xq,  ?/o),  and  a 
given  line: 

(a)  When  the  inclination  of  the  line, 
<l>,  and  the  coordinates  of  one  of  its 
points,  (xi,  yi),  are  given,  we  have 
from  the  figure : 

QP^z=z{xq  — Xi)  sin  </>— (2/0— 2/i)  cos  <^, 

(&)   When  the  equation  of  the  line  is 

given  in  the  form  Ax -\-  By  -{-  C  =  Q,  use  the  following  formula  rf 

Ax,  +Byo  +  C 


D 


Va^  +  B^ 
Here  the  vertical  bars  mean  '^  the  absolute  value  of.  '* 


*  Proof:    By  trigonometry,  tan  (02  —  <t>\) 


tan  <p2  —  tan  0i 


1-f-  tan  02  tan  0i 

t  Proof:  Show  that  the  foot  of  the  perpendicular  from  Po  to  the  line 
Ax  4-  /^j(  4-  C  =  0  has  the  coordinates  x^  =z  xq  —  hA,  t/2  =  2/0  —  hB,  where 
h=z{Axo  +  ByQ  +  C)  /  (A^  -f-  B^). 


CHAPTER  III. 

THE  CIRCLE :   EQUATIONS   OF  THE  FORM  X^  -{- y^  -^Dx  +  j&y  +  F  =  0. 

16.  The  equation  of  a  circle  is  the  algebraic  expression  of  the 
condition  which  x  and  y  must  satisfy  in  order  that  the  point  {x,y) 
shall  Ue  on  the  circle  (see  §  9  and  §  10). 


0 


X 


17.  To  find  the  equation  of  a  circle  when  its  radius,  r,  and  the 
coordinates  (a,  /9)  of  its  centre  are  given: 

When  the  centre  is  at  the  origin  (0,  0),  this 
equation  becomes 

x^+y^  =  r\ 

18.  The  equation  of  any  circle  is  of  the  fonn  x^-\-y^-\-Dx-\-Ey-\-F=zO. 
Conversely,  every  equation  of  the  fonn  x*  •^- y*  ■}-  Dx -\- Ey -\-  F  =0  can  be  re- 
duced to  the  fonn  (x  -f  -)» +  (y-\-  -^y  =  -(D'  +  ^  —  4F),  and  therefore  rep- 
resents a  circle  with  centre  at  ( —  Z)/2,  —  E/2),  or  a  single  point,  or  no  locus,  ac- 
cording as  2)2  -f-  £^  —  4F  is  positive,  zero,  or  negative.  When  we  say,  in  brief, 
that  the  locus  of  any  equation  of  the  fonn  x^  -{- y^  -\-  Dx  -\-  Ey  -}-  F  =Oisa, 
''circle,"  we  must  understand  that  the  "circle"  may  be  "real,"  "null,"  or 
"imaginary." 

19.  To  find  the  centre  and  radius  of  a  circle  whose  equation  is 
given,  do  not  use  a  formula,  but ' '  complete  the  squares"  of  the  terms 
in  X  and  y  in  each  case,  and  compare  with  the  standard  equation  in 
the  manner  just  indicated. 

20.  In  problems  concerning  tangents  to  a  circle,  use  the  fact  that 
the  tangent  is  perpendicular  to  the  radius  drawn  to  the  point  of  con- 
tact. 


52 


CHAPTER  IV. 
THE  parabola:  if  =  2px. 

21.  Definition:    The  locus  of  a  point  which  moves  so  that 

its  distance  from  a  fixed  point  _ 
its  distance  from  a  fixed  line     "" 

is  cdled  a  parabola. 

The  fixed  point  is  called  the  focus  and  the  fixed  line  the  directrix. 
The  line  perpendicular  to  the  directrix 
through  the  focus  is  called  the  principal 
axis.  There  is  evidently  only  one  point 
of  the  principal  axis  which  is  also  a  point 
of  the  curve,  namely  the  point  half  way 
between  the  focus  and  the  directrix;  this 
point  is  called  the  vertex. 

22.  If  we  take  the  vertex  as  the  ori- 
gin and  the  principal  axis  as  the  axis  of  x^ 
the  equation  of  the  parabola  is 

2/'  =  2px, 

where  p  =  the  distance  between  focus  and 
directrix.* 

23.  The  form  of  the  curve  is  therefore  that  shown  in  the 
figure,  t     By  definition  PF  =  PH  for  every  point  P  on  the  curve. 

The  breadth  of  the  curve  at  the  focus  is  called  the  lotus  rectum, 
and  is  equal  to  2p. 

*  Proof:  If  (x,  y)  is  any  point  on  the  curve,  then 

\/(x  —  ipy  +  (y-oy  =  x-hip. 

Many  British  authors  write  the  equation  in  the  form  y^  =  4aa;,  to  avoid 
fractions.  Other  writers  use  y^  =  4px  for  the  same  purpose ;  this  latter  form, 
however,  is  unfortunate,  since  2p  is  a  fairiy  well-established  notation  for  the  latus 
rectum  in  each  of  the  conies. 

t  Thus  when  xisO,  y  is  0.  When  x  increases,  y increases,  plus  and  minus; 
the  curve  is  symmetrical  with  respect  to  the  x-axis.  When  x  is  negative,  y  is 
imaginary.    When  x  =  p/2,  y  =:  ±  p;  when  x  =  2p,  y  =:±  2p, 


TT 

y 

^ 

M 

A' 

J) 

f 

(F          X. 

% 

\ 

n 

\ 

53 


54 


ANALYTIC   GEOMETBY. 


24.     To  find  the  equation  of  a  tangent  to  the  parabola  t/'  =  2'pXt 
use  one  of  the  following  formulas : 
(a)    When  the  point  of  contact,  {x^,  2/i) ,  is  given  :* 

yiy  =p(x  +  Xi); 
(h)     When  the  slope,  m,  is  given  :t 


y 


I      V 


A  line  perpendicular  to  a  tangent  at  the  point  of  contact  is 

called  a  normal. 

If  the  tangent  and  normal  at 

any    point   P   meet   the    principal 

axis  at  T  and  N,  the  projections 

of  PT  and  PN  on  the  principal  axis 

are  called  the  suhtangent  and  sub- 

normal,  respectively.  The  suhtan- 
gent is  bisected  by  the  vertex.  The 
subnormal  is  constant,  equal  to  the 

semi-latus  rectum,  p. 


25.  The  locus  of  the  middle  points  of  a 
set  of  parallel  chords  in  the  parabola  is  a 
straight  line  parallel  to  the  principal  axis;  such 
a  line  is  called  a  diameter.  In  the  parabola 
y^  =  2px,  if  the  slope  of  the  parallel  chords  is  m, 
then  the  equation  of  the  diameter  is  y  =z  p/m.X 


V/// 


X 


*  Proof:  Let  P2  =  {x\  +  A,  1/1  +  A;)  be  a  second  point  on  the  curve,  near  Pi; 
then  the  sloye  of  the  tangent  at  Pi  will  be  the  limit  of  k/h  as  P2  approaches  Pi 

along  the  curve,  namely  m  =  p/y-^.    Then  use  §  11.     The  slope  of  the  curve  may 
also  be  found  by  the  general  method  of  the  differential  calculus. 

t  Proof :  Determine  jS  so  that  the  line  1/  =  ma;  +  /3  shall  have  only  one 
point  in  common  with  the  curve.  [Eemember  that  a  quadratic  equation 
Ao^-\-Bx-\-C=:Q  will  have  equal  roots  if  5'  — 4^0  =  0.] 

%  Proof:  Let  {Xq,  y^)  be  any  point  of  the  required  locus;  find  the  points  of 
intersection  of  the  curve  and  a  line  through  {Xq,  2/0)  with  slope  m;  then 
express  the  condition  that  (a^o,  ?/o)  shall  be  the  middle  point  between  these  two 
points.  [Remember  that  the  sum  of  the  roots  of  a  quadratic  equation 
4a:'  4-  Bx  -f-  C  =  0  is  —  B/A.] 


anaIjYTic  geometry. 


66 


25a.  Among  the  many  properties  of  the  parabola  which  should  be  worked 
out  as  problems,  the  following  may  be  mentioned  as  especially  important,  and 
easy  to  remember: 


1.  The  normal  at  any  point  P  bisects  the  angle  formed 
by  the  line  from  P  to  the  focus  and  the  line  through  P 
parallel  to  the  principal  axis  (parabolic  mirror). 


2.  If  Pi,  P2, . . .  are  any  points  on  a  parabola, 
the  distances  of  these  points  from  the  principal 
axis  are  proportional  to  the  squares  of  their  dis- 
tances from  the  tangent  at  the  vertex. 


3.  If  the  tangents  at  P  and  Q  inter- 
sect at  T,  and  if  M  is  the  middle  point  of 
the  chord  PQ,  then  the  line  through  T 
and  M  is  a  diameter,  and  the  segment 
TM  is  bisected  by  its  point  of  intersec- 
tion with  the  curve. 


4.  The  locjiis  of  the  foot  of  the  perpendicular  from  the 
focus  on  a  moving  tangent  is  the  tangent  at  the  vertex. 


56 


ANALYTIC   GEOMETBY. 


5.  The  locus  of  the  point  of  intersection   of  perpen- 
dicular tangents  is  the  directrix. 


Note.    The  usual  methods  for  constructing  a  parabola  should  also  be  given. 


CHAPTER  Y. 
THE  ELLIPSE :  li^X^  +  a^  =  a*6'. 

26.  Definition  :    The  locus  of  a  point  which  moves  so  that 

its  distance  from  a  fixed  point  

its  distance  from  a  fixed  line 

(where  e  is  a  constant  less  than  1),  is  called  an  ellipse. 

The  fixed  point  is  called  the  focus,  the  fixed  line  the  directrix, 
and  the  constant,  e,  the  eccentricity.  The  line  perpendicular  to  the 
directrix  through  the  focus  is  called  the  principal  axis.  There  are 
evidently  two  points  of  the  principal  axis  which  are  also  points  of 
the  curve;  these  two  points  are  called  the  vertices,  and  the  point  half 
way  between  them  is  called  the  centre. 

27.  If  we  let  2a  =  the  distance      -^1   Yf  ^ ]L 

between  the  vertices,  then  :*  i    J^ — ^ — ->|^....^--..J 

the  distance  between  the  centre  and  either  vertex  is  a; 
the  distance  between  the  centre  and  the  focus  is  ae; 
the  distance  between  the  centre  and  the  directrix  is  a/e. 

28.  If  we  take  the  centre  as  the  origin  and  the  principal  axis 
as  the  axis  of  x,  the  equation  of  the  ellipse  is 

where  b  is  an  abbreviation  for  a  V^l  —  e^.-f     Note  that  b  <  a. 


and 


*  Proof:   Since  the  vertices,  V  and  F',  are  points  of  the  curve,  VF/VD  =  e 
V'F/V'D  =  e;  that  is. 


a-—CF  .      a  +  CF 

whence  CF  =  ae  and  CD  =  a/e. 

t  Proof:  If  {x,  y)  is  any  point  on  the  curve,  then 


V(a;  +  ae)='-f  (y— 0)'_^ 


- 
e 


57 


58 


ANALYTIC   GBOMETBY. 


29.  The  form  of  the  curve  is  therefore  that  shown  in  the 
figure.*  The  right  triangle  enables  us  to  find  any  one  of  the  three 
quantities  a,  b,  and  e,  when  the  other 
two  are  given. 

The  symmetry  of  the  equation 
shows  that  the  curve  might  equally 
well  have  been  obtained,  with  the  same 
eccentricity,  e,  from  a  second  focus  and 
directrix,  shown  on  the  right. 

The  breadth  of  the  curve  at  either 
focus  is  called  the  latics  rectum,  and  is 
equal  to  2a(l  —  e^),  or  2b'^/a. 


i+p  =  l. 


30.     Let  P  be  any  point  of  the  ellipse,  F  and  F'  the  foci,  and  PH 
and  PH'  the  perpendiculars  from  P  to  the  directrices ;  then 

(a)  PF/PH  =  e  and  PF'/PH'  =  e, 

by  definition  of  the  curve.     Further- 
more :t 


(6) 


PF  +  PF'  =  2a. 


In  fact,  the  ellipse  is  often  defined  as  the  locus  of  a  point  which 
moves  so  that  the  sum  of  its  distances  from  two  fixed  poinis  is  constant, 

31.  If  a  circle  be  described  upon  the  major  axis  of  an  elHpse 
as  diameter,  each  ordinate  in  the  ellipse  is  to 
the  corresponding  ordinate  in  the  circle  as  b  is 
to  a.i  In  fact,  the  ellipse  is  often  defined  as 
the  locus  of  points  dividing  the  ordinates  of  a  circle 
in  a  constant  ratio. 

From  this  property  it  follows  that  the  area  of 
an  ellipse  is  irab. 


*  Thus,  when  i/  =  0,  x  =  ±  a;  when  x  :=  0,  t/  =  ±  6.  The  curve  is  sym- 
metrical with  respect  to  both  axes.  In  first  quadrant,  as  x  increases,  y  decreases 
(slowly  when  x  is  small,  and  rapidly  when  x  approaches  a). 

t  Proof:  PF  =  e(PH)  and  PF'  =  e(PH'),  so  that  PF  +  PF'  =  eiHHf) 
=  c(2a/e)  =  2a. 

t Proof:  In  the  ellipse,  y  =  -  ^a'  —  x';  in  the  circle,  y  =  y/a'  —  a^\ 


ANALYTIC   GEOMETEY. 


59 


32.     To  find  the  equation  of  a  tangent  to  the  ellipse  -^  + 

use  one  of  the  following  formulas  :* 

(a)  When  the  point  of  contact,   (xi,  2/1),  is 

given : 


r- 


1. 


^2      i"       L2" 


1; 


(&)  When  the  slope,  m,  is  given: 
y  =z  mx  dz  s/a^m?  -\-  b^. 

33.  The  locus  of  the  middle  points  of  a  set  of  parallel  chords 
in  the  ellipse  is  a  straight  line  through  the  centre;  such  a  line  is 

x^      ^2 
called  a  diameter.    In  the  ellipse  ^  +  f^  =  1, 

if  the  slope  of  the  parallel  chords  is  m,  then 

the  slope  of  the  diameter  is r—  .* 

a^m 

Any  two  lines  through   the  centre,  such 

that  the  product  of  their  slopes  is  —  b^/a^,  are 

called  a  pair  of  conjugate  diameters,  because 

each  bisects  all  chords  parallel  to  the  other. 

34.  The  circle  described  in  §  31  is  called  the  auxiliary  circle. 
If  P  is  any  point  on  the  elHpse,  and   Q   the 

corresponding  point  on  the  auxiliary  circle 
(see  figure),  then  the  angle  <f>  which  CQ  makes 
with  the  axis  is  called  the  eccentric  angle  of 
the  point  P.     From  the  figure,  and  §  31, 

X  =z  a  cos(f>    and    y  =  b  sin^^ 

where  x,  y  are  the  coordinates  of  P. 

The  eccentric  angles  of  the  ends  of  two  conjugate  diameters 
differ  by  90°. 

*  Proof  as  in  the  case  of  the  parabola. 


60 


ANALYTIC   GEOMETBY. 


4a.   Among  the  many  properties  of  the  ellipse  that  should  be  worked 
out  as  problems,  the  following  are  especially  easy  to  remember: 


1.  The   normal   at   any  point  P  bisects  the   angle 
formed  by  the  lines  joining  P  with  the  foci. 


2.  The  locus  of  the  foot  of  the  perpendicular  from 
the  focus  on  a  moving  tangent  is  the  circle  on  the  major 
axis  as  diameter. 


3.  The  locus  of  the  point  of  intersection  of  per- 
pendicular tangents  is  a  circle  with  radius  Vo*  +  6*. 


4.  The  area  of  a  parallelogram  bounded  by  tan- 
gents parallel  to  conjugate  diameters  is  constant. 


Note.    The  usual  methods  for  constructing  an  ellipse  should  also  be  given. 


CHAPTER  VI. 

THE  HYPERBOLA :    6V  —  C?xf'  =  G?}?, 

35.    Definition:    The  locus  of  a  point  which  moves  so  that 

its  distance  from  a  fixed  point  

its  distance  from  a  fixed  fine 

(where  e  is  a  constant  greater  than  1),  is  called  a  hyperbola. 

The  fixed  point  is  called  the  focus,  the  fixed  line  the  directrix, 
and  the  constant,  e,  the  eccentricity.  The  line  perpendicular  to  the 
directrix  through  the  focus  is  called  the  'principal  axis.  There  are 
evidently  two  points  of  the  principal  axis  which  are  also  points  of 
the  curve;  these  two  points  are  called  the  vertices,  and  the  point 
half  way  between  them  is  called  the  centre. 


V  CD 


V  F^ 


36.  If  we   let   2a  =  the    distance       H ' — f "t^ 

between  the  vertices,  then:*  k—ij—- >k— ^-  x 

the  distance  between  the  centre  and  either  vertex  is  a; 
the  distance  between  the  centre   and  the  focits  is  ae; 
the  distance  between  the  centre  and  the  directrix  is  a/e. 

37.  If  we  take  the  centre  as  the  origin  and  the  principal  axis 
as  the  axis  of  x,  the  equation  of  the  hyperbola  is 

^  ^  ^'  -  1 
a'        b'  """.  ' 

where  b  is  an  abbreviation   for  aV^  —  l.f     Note  that  b  may  be 
greater  or  less  than  a,  or  equal  to  a. 

*  Proof:  Since  the  vertices,  V  and  V,  are  points  of  the  curve,  VF/VD  =s« 
and  F'F/F'i>  =  e;thatis, 

CF  —  a  ^      a-i-CF 

777c  =  e     and     — i—ftf.  =  «i 

a  —  CD  a  -f-  CD         ' 

whence  CF  =  ae  and  CD  =  a/e. 

t  Proof:  If  (x,  y)  is  any  point  on  the  curve,  then 


V(a;  —  aey  +  (y  —  0)'  __  ^ 

a 
X 

e 


61 


62 


ANALYTIC  GEOMETBY. 


,2  7,2  —  ^• 


38,  The  form  of  the  curve  is  therefore  that  shown  in  the  figure. * 
The  two  lines  through  the  centre 

with  slopes  ±  6/a  are  called  the 
asymptotes  of  the  hyperbola;  the 
two  branches  of  the  curve  ap- 
proach these  lines  more  and  more 
nearly  as  they  recede  from  the 
centre .  f  The  right  triangle  enables 
us  to  find  any  one  of  the  three 
quantities;  a,  b,  and  e,  when  the 
other  two  are  given. 

The  symmetry  of  the  equation 
shows  that  the  curve  might  equally 
well   have  been  obtained,  with   the   same   eccentricity,  e,  from  a 
second  jocus  and  directrix,  shown  on  the  left. 

The  breadth  of  the  curve  at  either  focus  is  called  the  latus  rectum, 
and  is  equal  to  2a(e^  —  1),  or  2b^/a. 

39.  Let  P  be  any  point  of  the  hyperbola,  F  and  F'  the  foci,  and 
PH  and   PH'  the     perpendiculars 
from  P  to  the  directrices ;  then 

(a)  PFjPH—e   and  PF'/PH'z=z  e, 

by  the  definition  of   the   curve. 
Furthermore :{ 

(6)      \PF—PF'\z=2a, 

In  fact,  the  hyperbola  is  often  defined  as  the  locus  of  a  point 
which  moves  so  that  the  difference  of  its  distances  from  two  fixed  points 
is  constant. 


*  Thus,  when  y  =  0,x=±  a;  when  rr  =  0,  or  ar  <  o,  t/  is  imaginary;  when  x 

increases  beyond  a,  y  increases,  plus  and  minus  (most  rapidly  when  x  is  little 

greater  than  a).    The  curve  is  symmetrical  with  respect  to  both  axes. 

V        b    1         o^  b 

t  For.    the   slope    -  =  -  -v/1 -,  approaches  -  as  a;  increases;   more- 

'         '  '^       X        a   1  X*  a 

over,  if  yi  is  the  ordinate  of  any  point  on  the  curve,  and  1/2  the  ordinate  of  the 

corresponding  point  on  the  asymptote,  then  the  difference  1/2  —  yi  approaches 

zero;  for  ys^  —  yi^  =  6',  and  therefore  1/2  —  t/i  =  67(2/2  +  2/1). 

t  Proof:    PF  =  e{PH)  and  PF  =  e{PH'),  so  that  [PF-^PF'i^  e{HW) 

8=  e  (2a/c)  =  2a. 


ANALYTIC   GEOMETRY. 


63 


40.  The  product  of  the  distances  from  any  point  of  a  hyperbola 
to  the  asymptotes  is  constant.  Hence,  the  hyperbola  is  often 
defined  as  the  locus  of  a  point  which 
moves  so  that  the  product  of  its  distances 
from  two  fixed  lines  is  constant.  (The 
distances  here  may  be  the  perpendicular 
distances;  or,  the  distance  to  each  line 
may  be  measured  parallel  to  the  other.) 

41.  An  important  special  case  is  that  of  the  "rectangular*! 
hyperbola,  whose   asymptotes  are  perpendicular 
(a  =  h) ;   the  equation  of   the  rectangular  hyper- 
bola  referred  to  its  asymptotes  as  axes  is  (by  §  40) 


xy 


42.     To   find    the   equation    of   a  tangent    to    the  hyperbola 
-2  —  l2=l;   ^6   o^^   ^^   *^®    following 

formulas  :* 

(a)  "When  the  point  of  contact,  {x^y  y^), 

is  given : 


XiX 


7>2    ^f 


(b)  "When  the  slope,  m,  is  given: 
y  =z  mx  ±  ^a^m^  —  b^. 

43.  The  locus  of  the  middle  points  of  a  set  of  parallel  chords 
in  the  hyperbola  is  a  straight  line  through  the  centre ;  such  a  line  is 
called    a    diameter.      In    the    hyperbola 

—  —  r;;  =  1,  if  the  slope  of  the  parallel 
a-^      o-^ 

chords  is  w,  the  slope  of  the  diameter 

will  be  ~* 
a^m 

Any  two  lines  through  the  centre, 

such  that  the  product  of  their  slopes  is  62/a2,  are  called  a  pair  of 

conjugate  diameters^  because  each  bisects  all  chords  parallel  to  the 

other. 


Proof  as  in  the  case  of  the  parabola. 


64 


ANALYTIC   GEOMETRY. 


43  a.  Among  the  properties  of  the  hyperbola,  the  following  are  easy  to 
remember : 


1.  If  a  line  cuts  the  hyperbola  and  its  asymptotes, 
the  parts  of  the  line  intercepted  between  the  curve  and 
the  asymptotes  are  equal.  In  particular,  the  portion 
of  any  tangent  intercepted  between  the  asymptotes 
is  bisected  by  the  point  of  tangency. 


2.  The  area  of  the  triangle  bounded  by  any  tan- 
gent and  the  asiymptotes  is  constant. 


Note.     The   usual   methods   of   constructing   a   hyperbola — especially   the 
rectangular  hyperbola — should  be  given. 


CHAPTER  VIL 

TRANSFORMATION  OF   COORDINATES.* 

44.  The  equation  of  a  curve  can  often  be  simplified  by  a 
**  change  of  axes,"  either  changing  to  a  new  origin  {xq,  i/o)> 
or  turning  the  axes  through  an  angle  6,  or  both. 

If  (x,  y),  {x\  y'),  {x" ,  y"),  are  the  coordinates  of  the  same 
point  Pj  referred  to  three  different  sets  of  axes,  as  in  the 
figures,  then 


T< 


4. 


a^*\\i 


"X-o 


« 
% 
% 

* 

A-- 

X 

' " 

X 

X  — '  Xq  "Y"  X 


X  =  x"  cos  $  —  y"  sin  d  t 
y  ==  x"  sin  B  +  y"  cos  6 


Suppose  now  that  the  point  P  is  allowed  to  move  under  cer- 
tain conditions  given  by  an  equation  in  x  and  y.  The  same 
condition  can  be  expressed  in  terms  of  x'  and  y'  or  in  terms 
of  x"  and  y"  by  substituting  in  the  given  equation  the  values 
of  X  and  y  just  found.  This  process  is  called  a  transforma- 
tion of  coordinates,  from  the  axes  x,  y  to  the  axes  x'y  y',  or  to 
the  axes  x'\  y" ;  and  the  new  equation  can  often  be  made  sim- 
pler than  the  given  equation  by  a  suitable  choice  of  Xq  and 
2/o,  or  e. 

*  See  also  the  chapter  on  polar  co-ordinates, 
t  These  last  formulas  are  most  easily  remembered  as  follows: 
X  =  easterly  displacement  of  P, 
=  (easterly  component  of  x")  -f-  (easterly  component  of  y") 
=  x"  cos  B  —  y"  sin  e\ 
y  =  northerly  displacement  of  P 
=  (northerly  component  of  x")  -}-  (northerly  component  of  3^') 
=  x"  sin  Q  +  y"  cos  B.  • 

65 


CHAPTER  VIII. 

GENERAL  EQUATION  OF  THE  SECOND  DEGREE  IN  X  AND  y, 

45.  The  general  equation  of  the  second  degree  in  x  and  y 
is  of  the  form 

Ax^  +  Bxy  +  Cy^^Dx  +  Ey  +  F  =  0. 

By  a  suitable  transformation  of  coordinates  this  equation  can 
always  be  brought  into  one  or  other  of  the  following  forms : 

A  V  +  CY  +  F'  =  0,     C'Y  +  D"x  =  0,     C'Y  +  F''  =  0, 

and  hence  can  be  shown  to  represent  a  conic  section,  using 
this  term  in  a  general  sense  to  include  (1)  an  ellipse,  which 
may  be  real,  null,  or  imaginary;  (2)  a  hyperbola,  or  a  pair 
of  intersecting  lines;  (3)  a  parabola,  or  a  pair  of  parallel  lines 
(distinct,  coincident,  or  imaginary).* 

The  student  should  be  able  to  plot  readily  the  locus  of  an 
equation  of  the  second  degree  in  any  of  the  simple  cases  men- 
tioned below — these  being  the  cases  which  occur  most  often 
in  practice. 

46.  To  plot  Ax^  +  Cy^  +  F  =  0,  where  A  and  C  have  the 
same  sign.  Find  the  intercepts  on  the  axes  (by  putting 
x  =  0  and  2/  =  0) ;  if  both  are  real,  we  have  an  ellipse  in  which 
a  =  the  larger  of  the  two  intercepts,  and  &  =  the  smaller;  or 
it  A  =  C,  the  ellipse  becomes  a  circle.  If  both  intercepts  are 
zero,  or  imaginary,  the  locus  is  a  single  point,  or  imaginary. 

*  Proof:  If  B^  —  4:AC  is  not  zero,  transform  to  parallel  axes  with 
origin  at  (Xo,  Vo),  and  choose  Xq  and  2/0  so  that  the  terms  of  the  first 
degree  in  the  new  equation  shall  vanish;  then  turn  the  axes  through  an 
angle  6,  and  choose  d  so  that  the  term  in  xy  shall  vanish.  If  JB'  —  4lAC 
=  0,  turn  the  axes  through  an  angle  d,  and  choose  6  so  that  the  term  in 
xy  shall  vanish;  then  transform  to  a  new  origin  {x^,  2/0),  and  choose  x^ 
and  2/0  so  that  the  constant  term  and  one  of  the  terms  of  the  first  degree, 
or  so  that  both  the  terms  of  the  first  degree  shall  vanish.  For  special 
methods  of  abbreviating  the  computation  in  numerical  cases  see  §  55, 
note.  • 


ANALYTIC   GEOMETBY.  67 

47.  To  plot  Ax^  +  Cy^  -\-  F  =  Of  where  A  and  C  have  oppo- 
site signs.  Unless  jP  =  0,  one  of  the  intercepts  will  be  real 
and  the  other  imaginary,  and  the  curve  will  be  a  hyperbola 
whose  principal  axis  is  the  axis  on  which  the  intercepts  are 
real.  To  find  the  slopes  of  the  asymptotes,  divide  by  x^  and 
find  the  limit  of  y/x  as  x  increases  indefinitely.  If  i^  =  0,  the 
locus  is  a  pair  of  intersecting  lines. 

48.  To  plot  Cy^  +  Dx  +  F  =  0.    Write  this  as 

This  is  a  parabola  with  vertex  at  Xq=^  —  F/D,  and  running 
out  along  the  positive  or  negative  axis  of  x.  Plotting  one  or 
two  points  will  fix  the  direction,  and  comparison  with  the 
equation  y^  ==  2px  will  give  the  semi-latus  rectum,  p. 

49.  To  plot  Ax^  +  Cy""  +  Dx  +  Ey  +  F  =  0.  Write  this  in 
the  form 

and  *' complete  the  squares'*;  then  reduce  to  the  form 
Ax'^  +  Cy'^  +  i^  =  0  by  an  obvious  change  of  origin. 

50.  To  ^lot  Cy""  +  Dx  +  Ey  +  F  =  0.  Complete  the  square 
of  the  terms  in  y  and  reduce  to  the  form  Cy^ -\-Dx-\-F  =  0 
by  an  obvious  change  of  origin. 

51.  To  plot  Bxy  -\-  F  =  0.  This  is  a  rectangular  hyberbola 
referred  to  its  asymptotes  (see  §41).  The  equation 
Bxy  +  Dx  +  Ey  -\-  F  =  0  can  be  reduced  to  this  form  by 
moving  the  origin  to  Xo  =  —  E/B,  2/0  =  —  D/B, 

52.  If  the  equation  to  be  plotted  does  not  come  under  any 
of  the  forms  just  considered,  a  fair  idea  of  the  position  of  the 
curve  may  be  found  by  the  following  very  elementary  method. 
Solving  the  equation  for  y  in  terms  of  x,  we  have,  if  C  is  not 
zero, 

where  X  is  an  expression  involving  x  alone.    Finding  the 


68  ANALYTIC   GEOMETRY. 

values  of  —  (Bx  +  E)/2C,  and  adding  and  subtracting  the 
values  of  V^/2C,  for  various  values  of  x,  we  can  find  as 
many  points  {x,  y)  on  the  curve  as  we  please.  Or,  again, 
solving  for  x  in  terms  of  y,  we  have,  if  A  is  not  zero, 

By^D        1 

2A      ^2J'^^- 

where  Y  is  an  expression  involving  y  alone.  From  this  equa- 
tion we  can  find  values  of  x  corresponding  to  as  many  values 
of  y  as  we  please. 

This  method  is  very  easy  to  remember,  but  does  not  give 
readily  the  exact  dimensions  of  the  curve. 

53.  The  center  of  the  curve  will  be  the  point  of  intersection 
of  the  two  lines 

2Ax  +  By  +  D  =  0,  ^ 
Bx  +  2Cy+E==0, 

except  when  B^ — 4AC  =  0,  in  which  case  these  lines  will  be 
parallel,  and  the  curve  has  no  center. 

54.  The  slopes  of  the  lines  joining  the  origin  with  the  infi- 
nitely distant  points  of  the  curve  (if  any)  are  given  by  writ- 
ing the  terms  of  the  second  degree  equal  to  zero :   • 

Ax^  +  Bxy  +  Cy^  =  0, 

dividing  through  by  x^  (or  2/^),  and  solving  for  y/x  (or  x/y), 

55.  If  a  more  detailed  discussion  of  the  curve  is  required, 
it  is  best  to  follow  the  special  methods  of  reduction  given  in 
the  text-books  (compare  foot-note  in  §45).t 

56.  The  student  should  be  familiar  with  the  geometric 
proof  that  all  the  **  conic  sections  "  can  be  obtained  as  plane 

*  The    student    of    the    calculus    will    recognize    these    equations    as 
dF/dx  =  0     and     dF/dy  =  0, 

F(x,  y)  ^Ax^  4-  Bxy  +  Cy"  +  Dx -\- Ey  +  F  =  0 

is  the  equation  of  the  curve. 

t  The  resulting  formulae  are  given  here  for  reference,  although  the 
problem  is  not  one  of  common  occurrence. 

Eequired,  to  plot  the  equation  Ax'  +  Bxy  +  Cy" -\- Dx  +  Ey  +  F  =  0. 


ANALYTIC   GEOMETBY.  69 

sections  of  a  right  circular  cone.  It  is  a  profitable  exercise 
to  construct  a  cone,  given  the  vertex  and  a  hyperbolic  section. 
It  should  also  be  made  thoroughly  clear  why  an  elliptic  sec- 
tion is  a  symmetrical  figure  instead  of  egg-shaped. 

Case  I.  Central  conic.  If  B'  —  4:AC  is  not  zero,  transform  to  the 
center  as  a  new  origin: 

x,=  (2CD  —  BE)/(B*  —  ^C),  y,=  {2AE  —  BD)/(B*  —  'iAC); 
then  turn  the  axes  through  a  positive  acute  angle  0  given  by 

tan  2d  =  B/iA  —  C). 
The  transformed  equation  will  be 

A'x'  +  C'y'  +  F'  =  0, 
where  F'  =  Dxo/2  +  Eyo/2  -f  F,  while  A'  and  C  are  found  by  solving 
the   equations   A'  +  C'  =  A  +  C,   A'  —  C'=z±  V{A  —  Cp^  5*,   where 
the  sign  before  the  radical  is  to  be  +  or  —  acording  as  B  is  positive 
or  negative.     The  reduced  equation  can  be  plotted  as  in  §§  46,  47. 

Case  II.  Parabolic  type.  If  J5'  —  4A.C  =  0,  the  equation  may  be 
written  in  the  form  (ax  +  cy)*  -\-  Dx  +  Ey  -^  F  =  0,  where  a  =  VA 
while  c==VC  or  c  =  —  VC  according  as  B  is  positive  or  negative.  The 
locus  will  be  of  the  parabolic  type.     Take  as  a  new  axis  of  x'  the  line 

ax  -{-  cy  +  in  =  0, 
where    m=  (aB-\- cE)/2(A -\- C),    and    choose    the    positive    direction 
along  this  line  so  that  it  shall  make  a  (positive  or  negative)  acute  angle 
with  the.  axis  of  x.    This  line  will  be  the  principal  axis  of  the  curve. 
Two  subcases  may  now  occur. 

(a)  If  a/c  is  not  equal  to  D/E,  take  as  axis  of  y'  the  line 
ex  —  ay  +  n  =0, 
where  n=(^  +  C)    (m*  —  F) / (aE  —  cD) .     This  line  will  be  the  tan- 
gent at  the  vertex,  and  the  transformed  equation  will  be 

y'^  =  2px', 
where  2p=(cP  —  aE)/V  (A  +  C)'.    The  locus  is  a  true  parabola, 
(d)  If  a/c  =  D/E,  the  equation  referred  to  the  axis  of  x'  will  be 
y''z=zm'  —  F, 
which  represents  a  pair  of  distinct,  coincident,   or  imaginary  parallel 
lines. 


CHAPTER  IX. 


SYSTEMS   OP   CONICS. 


57.  If  TJ  and  V  are  expressions  of  the  second  degree  in  x 
and  2/,  the  equations  V  =  0  and  y  =  0  will  represent  conies; 
then  (a)  the  equation  V  -{-kV  =  0,  where  k  is  any  constant, 
will  represent  another  conic  passing  through  all  the  points 
of  intersection  of  the  first  two,  and  having  no  other  points  in 
common  with  either  of  them;  and  (&)  the  equation  TJy  =  Q 
will  represent  a  curve  made  up  of  the  two  conies  TJ  =0  and 
Y=zO  taken  together.  Corresponding  theorems  hold  good  if  TJ 
and  y  are  any  expressions  in  x  and  y  (not  necessarily  of  the 
second  degree). 

58.  To  find  the  equation  of  a  conic  through  five  points,  let 
u  =  0  and  i;  ==  0  be  the  equations  of  the  lines  PJP^  ^^^  -^z^^^ 
and  let  u'  =  0  and  v'  =  0  be  the  equations  of  the  lines  P^P^ 
and  P2-P4.  Then  uv  +  ku'v'  =  0,  where  k  is  any  constant, 
will  be  the  equation  of  a  conic  through  these  four  points.  It 
remains  to  determine  k  so  that  this  conic  shall  pass  through  P^. 

59.  The  equation 


2   I    7,      ^^ 


where  k  is  an  arbitrary  constant,  represents  a  family  of  con- 
focal  ellipses  and  hyperbolas,  which  intersect  at  right  angles. 


70 


CHAPTER  X. 


POLAR  COORDINATES. 

This  chapter,  placed  here  for  convenience  of  reference,  may 
well  be  introduced,  in  teaching,  much  earlier  in  the  course. 

60.  It  is  often  convenient  to  represent  the  position  of  a 
point  P  by  giving  the  angle,  <^,  which  the  line  through  0  and 
P  makes  with  the  a^-axis,  and  the  distance,  r,  from  0  to  P  along 
this  line.  The  angle  <j>  is  called  the  vectorial  angle,  or  simply 
the  angle,  of  the  point  P,  and  is  measured  from  the  positive 
direction  of  the  axis  of  x  to  the  positive  direction  of  the  line 
through  0  and  P.  The  distance  r  =  OP  is  called  the  radius 
vector  of  the  point  P,  and  is  positive  or  negative  according  as 
it  runs  forward  or  backward  along  the  line  through  0  and  P. 

It  is  customary  to  take  r  positive,  and  let  <^  range  from  0° 
to  360°. 

61.  From  the  figure, 

x  =  r  cos  <f>,    y  =  r  sin  <^, 

x^  -\'y^  =  r^,     y/x  =  tan  <j>. 

By  the  aid  of  these  relations,  we  can  trans- 
form any  equation  from  rectangular  to 
polar  coordinates,  and  vice  versa. 

62.  The  polar  equation  of  a  conic,  referred  to  the  focus  as 
origin,  and  the  principal  axis  as  axis  of  x 

(see  figure)  is 


r  = 


P 


1  —  e  cos  6  ' 


^  .. 

J 

^. 

r\ 

O 

where  p  is  the  semi-latus  rectum,  and  e  the 
eccentricity. 

63.  Plotting  curves  in  polar  coordinates  is  an  excellent 
exercise  in  reviewing  the  trigonometric  functions.  The  work 
should  be  so  arranged  that  no  critical  value  of  the  function 
occurs  between  two  successive  assigned  values  of  0, 

71 


CHAPTER    XI. 

Coordinates  in  Space. 

64.  Four  methods  are  in  use  for  representing  numerically 
the  position  of  a  point  in  space.  If  Ox,  Oy,  Oz  are  three  mu- 
tually perpendicular  axes,  the  position  of  any  point  P  may 
be  determined  by : 

(1)  Rectangular  coordinates,  x^  y^  z; 

(2)  Polar  coordinates  in  space,  r,  a,  (3,  y,  where  the  angles 
a,  /3,  y  are  subject  to  the  restriction  cos^  a  -\-  cos^  p  +  ^^s^  y  =  1 ; 

(3)  Spherical  coordinates,  r,  <f>,  0,  where  <^=the  latitude  of 
P,  and  6  its  longitude : 

(4)  Cylindrical  coordinates  p,  B,  z. 

The  relations  between  the  various  sets  of  coordinates  are  as 
follows : 


x  =  r  cos  a, 
2/  =  r  cos  p, 
z  =r  cos  y. 


X  =  r  cos  <^  cos  ^      p^r  cos  </», 
y  =  rcos<l>sm6     p^  =  x^  -\-y^j 


z  =  r  sin  <f> 


+  2/^  +  s^  ==  r^ 


As  there  is  no  well-established  uniformity  in  the  use  of  the 
letters  in  spherical  coordinates,  or  in  the  choice  of  the  positive 
directions  along  the  axes,  it  is  important,  in  reading  any 

72 


ANALYTIC   GEOMETBY. 


73 


author,  to  note,  on  a  figure,  the  exact  meanings  of  the  letters 
he  employs. 

65.  Distance  between  two  points,  in  terms  of  their  coordi- 
nates : 


PxPi  =  V(a^i  —  ^2)'  +  (2/1  —  2/2)'  +  («i  —  ^2)'' 


i 

/ 

U 

7 

.^'' 

v 

^''           / 

7 

0 

y 

/ 

/ 

/ 

/        / 

^             / 

/ 

66.  Angle  ^  between  two  lines  whose  direction  cosines  are 
given : 

cos  ^  =  1^2  +  ^1^2  +  Wl^2» 

where  l^  =  cos  a^,  m^  =  cos  p^,  tij  =  cos  yi,  etc.* 

67.  Equation  of  a  plane: 

Ix  +  my  -\-  nz  =  p, 

where  p  =  perpendicular  distance  from 
the  origin,  and  I,  m,  w  =  the  direction 
cosines  of  the  normal  to  the  plane,  t 

Every  equation  of  the  form 
Ax  -{-  By  -\-  Cz  -{-  D  =  0  represents  a 
plane ;  for,  it  can  be  thrown  into  the  form* 
lx-\-my  -\-nz  =  p  by  dividing  through  by  ^/A^-\-B^-\-C^. 

♦Proof:  let  (1)  and  (2)  be  lines  through  the  origin,  parallel  to  the 
given  lines;  on  these  lines  take  points  Pi,  P,  at  a  distance  r  from  the 
origin;  then 
F^F^^  =  r*  +  r»  —  2rr  cos  ^  =  (rZ^  —  rl^y  +  {rm^  —  rm,)«  +  (rrii  —  m»)*. 

t  Proof :  The  foot  of  the  perpendicular  is  ^  =  {pi,  pm,  pn)  ;  take  N' 
=  {2pl,  2pm,  2pn)  and  express  the  condition  that  the  point  (x,  y,  e) 
shall  be  equidistant  from  0  and  N'. 


74  ANALYTIC   GEOMETRY. 

68.  Equation  of  sphere  with  center  at  the  origin : 

69.  Equation  of  ellipsoid,  with  center  at  the  origin : 

x^a^  +  2/V&=*  +  z^c^  =  1. 

70.  Any  equation  in  x,  y,  z  will  represent  a  surface  (real 
or  imaginary),  the  form  of  which  can  be  investigated  by  the 
method  of  plane  sections.  Thus,  putting  x  =  Xj,  the  equation 
becomes  an  equation  in  y  and  z,  which  represents  a  curve  in 
the  plane  x  =  x^]  similarly  for  y=  y^  and  z  =  z^. 

71.  Any  equation  of  the  second  degree  in  x,  y,  z  represents 
a  (real  or  imaginary)  surface  of  the  second  degree,  or  coni- 
coid.     The  types  of  real  conicoids  are  as  follows: 

(1)  Ellipsoid,  with  semi-axes  a,  h,  c.  Special  case:  ellip- 
soid of  revolution,  generated  by  rotating  an  ellipse  about  its 
major  axis  (prolate  spheroid)  or  about  its  minor  axis  (oblate 
spheroid). 

(2)  Eyperholoid  of  two  sheets.  Special  case:  generated 
by  rotating  a  hyperbola  about  its  principal  axis. 

(3)  Eyperholoid  of  one  sheet,  or  ruled  hyperholoid.  Spe- 
cial case :  generated  by  rotating  a  hyperbola  about  its  conju- 
gate axis.  Two  sets  of  straight  lines  can  be  drawn  on  this 
surface. 

(4)  Elliptic  paraboloid.  Special  case:  generated  by  rotat- 
ing a  parabola  about  its  principal  axis. 

(5)  Hyperbolic  paraboloid,  or  ruled  paraboloid.  A  saddle- 
shaped  figure,  on  which  two  sets  of  straight  lines  can  be 
drawn. 

(6)  Cone,  generated  by  a  straight  line  always  passing 
through  a  fixed  point  called  the  vertex,  and  always  touching  a 
fixed  conic,  called  the  directrix.  If  the  directrix  is  a  circle, 
the  cone  is  a  circular  cone  (right  or  oblique).  If  the  vertex 
recedes  to  infinity,  the  cone  becomes  a  cylinder.  On  any  cone 
a  single  set  of  straight  lines  can  be  drawn. 

The  student  should  become  familiar  with  at  least  the  shapes 
of  these  surfaces,  through  diagrams  or  models. 

Any  plane  section  of  any  surface  of  the  second  degree  is  a 
conic. 


A  SYLLABUS  OF  DIFFERENTIAL  AND 
INTEGRAL  CALCULUS. 

This  syllabus  is  intended  to  include  those  facts  and  methods  of  the 
calculus  which  every  student  who  has  completed  an  elementary  course 
in  the  subject  should  have  so  firmly  fixed  in  his  memory  that  he  will 
never  think  of  looking  them  up  in  a  book.  The  topics  here  mentioned 
are  therefore  not  by  any  means  the  only  topics  that  should  be  included 
in  a  course  of  study,  nor  does  the  arrangement  of  these  topics,  as 
classified  in  the  following  table  of  contents,  necessarily  indicate  the 
order  in  which  they  should  be  presented  to  a  beginner. 

Table  op  Contents. 

part  i.    functions  of  a  single  variable. 

Chapter  I.    Functions  and  their  Graphical  Bepresentation. 
Function  and  argument.     Tables.     Graphs. 
The  elementary  mathematical  functions. 
Continuity. 

To  find  a  mathematical  function  to  represent  an  empirically  given 
curve. 

Chapter  II.    Differentiation,    Rate  of  Change  of  a  Function. 

A.  Definitions  and  notation.  Rate  of  change  of  a  function,  or  slope 
of  the  curve.  Derivatives.  Increments  and  differentials.  Higher 
derivatives. 

B.  To  find  the  derivative  when  the  function  is  given. 

Rules  for  differentiating  the  elementary  functions.  Differentiation 
of  implicit  functions,  and  of  functions  expressed  in  terms  of  a  param- 
eter. 

C.  To  find  the  derivative  when  the  function  itself  is  not  given;  set- 
ting up  a  differential  equation. 

Useful  theorems  on  infinitesimals. 

D.  Applications  of  differentiation  in  studying  the  properties  of  a 
given  function.  Slope;  concavity;  points  of  inflexion.  Maxima  and 
minima.     Multiple  roots.     Small  errors. 

Chapter  III.     Integration  as  Anti-Differentiation.     Simple  Dif- 
ferential Equations. 
Definition  of  an  integral.     Constant  of  integration. 
Formal  work  in  integration.     Use  of  tables  of  integrals.     Method  of 
substitution,  and  method  of  integration  by  parts. 
Simple  differential  equations. 

75 


76  CALCULUS. 

Chapter    IV.     Integration   as    the   Limit   of   a    Sum.     Dkpiniti 
Integrals. 
Definition  of  the  definite  integral  of  f(x)dx  from  a:  =  a  to  a:  =  5. 
Fundamental  theorem  on  the  evaluation  of  a  definite  integral. 
Duhamel's  theorem. 

Approximate   methods    of   integration:    squared   paper;    Simpson's 
rule;  the  planimeter;  expansion  in  series. 

Definite  integral  as  a  function  of  its  upper  limit. 

Chapter  V.    Applications  to  Algebra:   Expansion  in  Series;   In- 
determinate Forms. 
Taylor's    theorem    with    remainder.       Maelaurin's    theorem.       Im- 
portant series. 

Theorem  on  indeterminate  forms. 

Chapter  VI.    Applications  to  Geometry  and  Mechanics. 
TangeAt  and  normal.     Subtangent  and  subnormal. 
Differential  of  arc  (in  rectangular  and  polar  coordinates). 
Eadius  of  curvature. 
Velocity  and  acceleration  in  a  plane  curve. 

PART  II.     functions  OF  TWO  OR  MORE  VARIABLES.* 

•  In  preparation. 


CHAPTER     I. 

Functions  and  Their  Graphical  Representation. 

1.  Function  and  argument. — In  many  problems  in  prac- 
tical life  we  have  to  deal  with  the  relation  between  two  variable 
quantities,  one  of  which  depends  on  the  other  for  its  value. 

For  example,  the  temperature  of  a  fever  patient  depends  on  the  time; 
the  velocity  acquired  by  a  falling  body  depends  on  the  distance  fallen; 
the  weight  of  an  iron  ball  depends  on  its  diameter,  etc. 

In  general,  if  any  quantity  y  depends  on  another  quantity 
Xj  then  y  is  called  a  function  of  x,  written,  for  brevity, 
y  =  f{x)f  and  the  independent  variable  x  is  called  the  argu- 
ment of  the  function.  More  precisely  stated,  the  notation 
y  =  f{x)  means  that  to  every  value  of  the  argument  x  (within 
the  range  considered),  there  corresponds  some  definite  value 
of  the  function,  t/;  the  value  of  y,  or  /(x),  corresponding  to 
any  particular  value  ii;  =  a  is  denoted  by  /(a). 

If  several  values  of  y  correspond  to  each  value  of  x,  we  have  what 
is  called  a  *' multiple  valued  function  of  x,"  which  is  really  a  collection 
of  several  distinct  functions.  For  example,  if  y^=zx,  then  2/  =  ±  y/x, 
which  is  a  double  valued  function  of  x. 

Any  mathematical  expression  involving  a  variable  re  is  a 
function  of  x ;  but  there  are  many  important  functional  rela- 
tions which  cannot  be  expressed  in  any  simple  mathematical 
form. 

2.  A  function  is  said  to  be  tabulated  when  values  of  the 
argument  (as  many  as  we  please,  preferably  at  regular  inter- 
vals) are  set  down  in  one  column,  and  the  corresponding 
values  of  the  function  are  set  down  in  another  column,  op- 
posite the  first.  For  example,  in  a  table  of  sines,  the  angle 
is  the  argument,  and  the  sine  of  the  angle  is  the  function. 

3.  A  function  may  also  be  exhibited  graphically,  as  follows : 
Lay  off  the  values  of  the  argument  as  abscissas  along  a  (hori- 
zontal) axis.  Ox,  and  at  each  point  of  the  axis  erect  an  ordi- 

6  77 


78  CALCULUS. 

nate,  y,  whose  length  shall  indicate  the  value  of  the  function 
at  that  point;  a  curve  drawn  through  the  tops  of  these  ordi- 
nates  is  called  the  curve,  or  the  graph,  of  the  function.  It 
should  be  clearly  understood,  however,  that  it  is  the  height  of 
the  ordinate  up  to  the  curve,  rather  than  the  curve  itself,  that 
represents  the  function. 

In  plotting  the  curve  for  any  function,  it  is  important  to 
indicate  on  each  axis  the  scale  which  is  used  on  that  axis,  and 
the  name  of  the  unit.  For  example,  if  we  plot  distance  as 
a  function  of  the  time,  the  units  on  the  2/-axis  may  represent 
feet,  and  those  on  the  x-axis,  seconds.* 

The  obvious  method  of  obtaining  the  graph  of  the  sum  or  difference  of 
two  functions  directly  from  the  graphs  of  those  functions  should  be 
noted. 

4.  The  elementary  mathematical  functions. — In  many  im- 
portant cases,  the  relation  between  the  function  and  the  argu- 
ment can  be  expressed  by  a  simple  mathematical  formula. 
For  example,  if  5  =  the  distance  fallen  from  rest  in  the  time  t, 
then  s  =  igt^.  In  such  cases,  the  value  of  the  function  for 
any  given  value  of  the  argument  can  be  found  by  simple  sub- 
stitution in  the  formula. 

The  most  important  elementary  mathematical  functions  are 
the  following: 

Algebraic  functions:  ex,  c/x\  x^,  x^;  "s/x  {x  positive). 

Here  V^  =  the  positive  value  of  y  for  which  y^  =  x. 

Trigonometric  functions:  sin  x,  cos  x,  tan  x  {x  in  radians). 

Exponential  function:  e*  (e  =  2.718  .  .  .). 

Logarithmic  function:  loge  x  {x  positive). 

The  student  should  be  thoroughly  familiar  with  the  curves 
of  each  of  these  functions,  so  as  to  be  able  to  sketch  them,  or 
visualize  them,  at  any  moment;  many  of  the  essential  prop- 
erties of  the  functions  can  be  obtained  by  inspection  of  the 
curve. 

•  It  is  not  necessary  that  the  lengths  representing  the  units  of  x  and  y 
shall  be  equal;  scales  should  be  so  chosen  that  the  completed  graph  is  of 
convenient  size  to  fit  the  paper.  In  applications  to  geometry,  however 
(see  Chapter  VI),  the  scales  must  be  equal. 


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78d 


CALCULUS.  79 

He  should  also  be  familiar  with  the  formulas  necessary  for 
handling  expressions  involving  these  functions.  The  better 
drilled  the  student  is  in  this  formal  algebraic  work,  the  more 
rapid  progress  can  he  make  in  the  really  vital  parts  of  the  sub- 
ject. (See  chapters  of  this  report  on  algebra  and  trigo- 
nometry.) 

5.  Next  in  importance  are  the  following:  the  hyperbolic 
functions,  which  are  coming  more  and  more  into  use : 

sinh  x={e''  —  e-'^) /2,     cosh  x={e^-\-  e"*) /2, 
tanh  x  =  (e*  —  e-^)/(e'^  -f  e"*) ; 
the  inverse  trigonometric  functions: 

sin-^  X  =  the  angle  between  —  7r/2  and  +  V2  radians 

(inclusive)  whose  sine  is  aj;* 
cos"^  X  =  the  angle  between  0  and  ir 

(inclusive)  whose  cosine  is  x; 
tan"^  X  =  the  angle  between  —  7r/2  and  +  ir/2 

(inclusive)  whose  tangent  is  x; 
and  the  inverse  hyperbolic  functions: 

sinh"^  X  =  the  value  of  y  for  which  sinh  y  =  x'y 

cosh"^  ic  ==  the  positive  value  of  y  for  which  cosh  y==x; 

tanh"^  X  =  the  value  of  y  for  which  tanh  y  =  x. 

It  should  be  noticed  that  the  curves  for  the  inverse  functions 
can  be  obtained  from  the  curves  for  the  direct  functions  by 
rotating  the  plane  through  180°  about  the  line  bisecting  the 
first  quadrant. 

Formulas  for  the  hyperbolic  functions  resemble  those 
for  the  trigonometric  functions,  but  the  differences  are  so 

*  The  symbol  sin"*  x  is  often  defined  as  simply  *  *  the  angle  whose  sine 
is  a;  *';  but  since  there  are  many  such  angles,  it  is  necessary  to  specify 
which  one  is  to  be  taken  as  **  the  "  angle,  if  the  symbol  is  to  have  any 
definite  meaning.  Thus,  if  sin  x=i^,  x  may  equal  7r/6,  or  Stt/G,  etc. ; 
but  only  one  of  these  values,  namely  7r/6,  is  properly  denoted  by  the 
symbol  sin"*  I.  Similarly  for  cos"^  x  and  tan"^  X',  and  also  for  cosh"*  x, 
which  is  like  y/x  in  this  respect.  The  conventions  adopted  to  avoid  am- 
biguity may  be  readily  recalled  from  the  figure,  if  we  note  that  in  each 
case  the  complete  curve  consists  of  two  or  more  *' branches, ' '  and  that 
that  one  is  taken  as  the  *' principal  branch"  which  passes  through  the 
origin,  or  which  lies  nearest  the  origin  on  the  positive  side  of  the  a;-a3di 


so  CALCULUS. 

confusing  that  it  is  better  not  to  try  to  memorize  any  formulas 
for  the  hyperbolic  functions,  but  to  look  them  up  whenever 
they  are  needed.  (The  list  in  B.  0.  Peirce's  Table  of  Integrals, 
for  example,  is  entirely  adequate.) 

6.  Continuity.  A  function  y  =  f{x)  is  said  to  be  continu- 
ous at  a  given  point  x  =  a^  if  a  small  change  in  x  produces 
only  a  small  change  in  y;  or  more  precisely,  if  f{x)  always  ap- 
proaches f{a)  as  a  limit  when  x  approaches  a  in  any  manner. 

A  function  may  be  cZiscontinuous  at  a  given  point  in  three 
ways:  (1)  it  may  become  infinite  at  that  point,  as  y=i\/x  at 
a;  =  0;  or  (2)  it  may  make  a  finite  jump,  as  i/^tan-^  (lA)  at 
^__Q.*  Qj.  ^3)  ^jjQ  limit  Jjf{x)  may  fail  to  exist  because  of  the 

oscillation  of  the  function  in  the  neighborhood  of  x  =  a,  as 
y  =  sin  1/x  at  x  =  0.  In  each  of  these  cases,  the  function  is, 
properly  speaking,  not  defined  at  the  point  in  question. 

A  good  example  of  a  discontinuous  function  is  the  velocity  of  a 
shadow  cast  by  a  moving  object  on  a  zig-zag  fence. 

In  what  follows,  we  shall  confine  our  attention  to  functions 
that  are  continuous,  or  that  have  only  isolated  points  of  dis- 
continuity. 

7.  To  find  a  mathematical  function  to  represent  an  em- 
pirically given  curve. — In  many  cases  the  form  of  the  func- 
tion is  given  only  empirically ;  that  is,  the  values  of  the  func- 
tion for  certain  special  values  of  the  argument  are  given  by 
experiment,  and  the  intermediate  values  are  not  accurately 


known  (for  example,  the  temperature  of  a  fever  patient,  taken 
every  hour).  In  such  cases,  the  methods  of  the  calculus  are 
not  of  much  assistance,  unless  some  simple  mathematical  law 
can  be  found  which  represents  the  function  sufficiently  accu- 

*  This  function  approaches  7r/2  when  x  approaches  0  from  above,  and 
—  v/2  when  x  approaches  0  from  below. 


CALCULUS.  81 

rately.*  This  problem  of  finding  a  mathematical  function 
whose  graph  shall  pass  through  a  series  of  empirically  given 
points  is  a  very  important  one,  which  is  much  neglected  in  the 
current  text-books.  The  complete  discussion  of  the  problem 
involves,  it  is  true,  the  theory  of  least  squares,  which  would 
undoubtedly  be  out  of  place  in  a  first  course  in  the  calculus ; 
but  an  elementary  treatment  of  the  problem  in  simple  cases 
would  be  very  desirable.f 

The  curves  which  are  most  likely  to  be  worth  trying,  in  any 
given  case,  are  these : 

y=za-\-'bx  (straight  line) ; 

y  =  a -{- hx -\- cx^  (parabola) ; 

y  ==  a -\- c/ {x -\- h)  (hyperbola); 

y  =  a  sin  (hx  +  c)  (sine  curve) ;  and 

y  =  ax"*. 

In  testing  this  last  curve,  put  y'  =  log  y,  x'  =  \og  x,  and 
a'  =  log  a,  and  see  whether  y'  and  x'  satisfy  the  straight  line 
relation  y'  =  a'-{-mx';  the  use  of  "logarithmic  squared 
paper  '  *  greatly  facilitates  the  process. 

The  student  should  be  familiar  with  all  the  possible  forms 
of  these  curves,  for  various  values  of  the  constants  a,  b,  c, 
and  m. 

*  If  no  simple  law  can  be  found  to  represent  the  entire  curve,  it  is 
sometimes  possible  to  break  up  the  curve  into  parts,  and  find  a  separate 
law  for  each  part. 

t  Numerous  examples  may  be  found  in  John  Periy's  '*  Practical 
Mathematics,'*  and  in  F.  M.  Saxelby's  ** Practical  Mathematics*' 
(Longmans,  1905). 


CHAPTER   11. 
Differentiation.     Rate  of  Change  of  a  Function. 

For  the  sake  of  clearness,  this  chapter  is  divided  into  four  parts, 
A,  B,  C,  D. 

A.    Definitions  and  Notation. 
8.  Rate  of  change  of  function;  slope  of  curve. — Given  a 

function,  yz=f{x),  one  of  the  most  important  questions  we 
can  ask  about  it  is,  what  is  the  rate  of  change  of  the  function 
at  a  given  instant? 

For  example,  the  distance  of  a  railroad  train  from  the  starting  point 
is  a  function  of  the  time  elapsed,  and  we  may  ask,  what  is  the  rate  of 
change  of  this  distance?  The  answer  is,  so-and-so  many  miles  'per  hour. 
Again,  the  volume  of  a  metal  sphere  is  a  function  of  the  temperature, 
and  we  may  ask,  what  is  the  rate  of  change  of  this  volume?  The  answer 
is,  so-and-so  many  cubic  inches  per  degree. 

If  the  graph  of  the  function  is  a  straight  line,  then  clearly 
the  rate  of  change  of  the  function  will  be  constant;  for,  at 


any  instant,  (change  in  2/) /(change  in  x)  =the  slope  of  the 
line. 

If  the  scales  along  x  and  y  are  the  same,  the  slope  of  the  line  =  tan  <f>, 
where  <f>  is  the  angle  which  the  line  makes  with  the  x  axis.  If  the 
scales  are  not  the  same,  the  slope  of  the  line  may  still  be  interpreted  as 
the  ratio  of  the  "side  opposite *'  to  the  '*side  adjacent'*  in  the  triangle 
of  reference  for  <f),  provided  each  side  is  measured  in  the  proper  units. 
For  example,  in  the  figure,  slope  =  7/3. 

82 


CALCULUS.  83 

If  the  graph  is  not  a  straight  line,  the  meaning  of  **rate 
of  change"  at  a  given  instant  must  be  made  more  precise,  as 
follows:  Consider  a  particular  value,  x  =  Xq',  give  x  an  arbi 
trary  change.  Arc,  and  compute  the  corresponding  change  in 
y,  namely,  A2/  =  /(Xo  +  Ao;) — fix^).  Then  the  ratio  Aiz/Arc 
may  be  called  the  average  rate  of  change  of  the  function  dur- 
ing the  interval  from  x  =  Xq  to  x  =  Xq-^  Lx.  (Geometrically, 
Ai//Aa;  is  the  slope  of  the  secant  FQ  in  the  figure.)  Now  let  ^x 
approach  zero,  so  that  the  interval  in  question  closes  down 
about  the  point  x  =  x^.  Then  the  ratio  Ly/^x  will  in  general 
approach  a  definite  limits  and  this  limit  is  called  the  actual 
rate  of  change  at  the  point  x  =  Xq.  (Geometrically,  the  limit 
of  Ay /Ax  is  the  slope  of  the  tangent  at  P.*) 

9.  Derivatives.  The  rate  of  change  of  a  function  y  =  f{x) 
at  any  point,  or  the  slope  of  the  curve  at  that  point,  is  called 
the  derivative  of  the  function  at  that  point,  and  is  denoted  by 
f'{x),  or  Da,y,  or  y\ 

The  notation  y  is  also  used,  but  only  when  the  independent 
variable  is  the  time. 

This  definition  of  the  derivative  of  a  function  as  the  limit 
of  Ay/ Ax  is  the  fundamental  concept  of  the  differential  cal- 
culus. It  is  desirable  that  the  meaning  of  the  definition  be 
made  perfectly  clear,  by  numerous  and  varied  illustrations, 
before  any  formal  work  in  differentiation  is  taken  up. 

10.  Increments  and  Differentials. — The  value  Ax  is  called 
the  increment  given  to  x,  and  Ay  the  corresponding  increment 

*  The  sense  in  which  the  tangent  line  is  the  "limit"  of  the  secant 
lines  should  be  made  thoroughly  clear.  First,  the  tangent  is  a  fixed 
line;  secondly,  the  secant  is  a  variable  line,  depending  on  the  value 
given  to  Ax  (that  is,  for  every  value  of  Ax,  except  the  value  0,  there 
^•7  a  corresponding  position  of  the  secant)  ;  thirdly,  the  angle  between  the 
tangent  and  the  secant  can  be  made  to  become  and  remain  as  small  as  we 
please  by  taking  Ax  sufficiently  small.  The  tangent  line  itself  does  not 
in  general  belong  to  the  series  of  secant  lines;  it  is  not  in  any  sense  the 
*'last  one'*  of  the  secants;  it  is  a  separate  line,  which  bears  a  special 
relation  to  the  series  of  secant  lines,  as  described.  The  student  may 
readily  convince  himself  that  the  tangent  is  the  only  line  through  P 
that  has  the  property  just  stated. 


84 


CALCULUS. 


produced  in  y.  The  value  that  ^y  would  have  if  the  curve 
coincided  with  its  tangent  (see  figure)  is  called  the  differential 
of  y  and  is  denoted  by  dy. 


s 

^ 

J 

Ax 

X 

:c 

X.+&X 

In  case  of  the  independent  variable  x^  the  differential  of  x 
is,  by  definition,  the  same  as  the  increment :  dx  =  Ax. 

The  use  of  differentials  gives  us  a  new  notation  for  the  de- 
rivativey 

/'(^)  =  l- 

Both  these  notations  are  in  common  use. 

Notice  that  Ay  and  dy  are  both  variables  which  approach 
zero  when  we  make  Ax  approach  zero;  dy/dx  is  a  constant, 
equal  to  tan  </>;  Ay /Ax  is  a  variable,  approaching  tan  ^  as  a 
limit.    Hence  we  may  write : 


and 


limg=r(x)  =  D,y  = 


dx 


tan  <^, 


dy  =  f(x)dx. 


These  relations  between  increments,  differentials,  and  deriva- 
tives should  be  thoroughly  mastered ;  they  are  readily  recalled 
by  the  figure.  Note  especially  that  Ax  and  dx  are  quantities 
measured  in  the  same  unit  as  x;  and  Ay  and  dy  in  the  same 
unit  as  y;  while  the  derivative,  dy/dx,  that  is,  the  slope,  is 
(in  general)  measured  in  a  compound  unit  (like  miles  per 
hour) . 


CALCULUS.  85 

If  the  lengths  representing  the  units  of  x  and  y  are  not  equal,  the 
slope  of  the  curve,  or  tan  0,  must  be  understood  in  the  generalized  sense 
explained  above. 

The  process  of  finding  the  derivative,  or  the  equivalent 
process  of  finding  the  differential  of  the  function  in  terms  of 
the  differential  of  the  argument,  is  called  differentiation, 

11.  Higher  derivatives.  Since  the  slope  of  the  curve  varies, 
in  general,  from  point  to  point,  the  derivative,  f{x),  is  itself 
a  function  of  x  (often  called  the  derived  function) ;  the  de- 
rivative of  f(x)  is  called  the  second  derivative  of  the  given 
function,  and  is  denoted  by  f'{x),  or  Dx^y,  or  y"  (or  by  y  in 
case  the  independent  variable  is  the  time) ;  and  so  on  for  the 
higher  derivatives. 

It  is  also  easy  to  define  second,  third,  .  .  .  differentials,  but 

they  are  not  of  great  importance.     One  matter  of  notation, 

however,  should  be  carefully  noticed,  namely  that  d^y/dx^  is 

d(dy/dx) 
commonly  used  to  denote  f"  (ic),  that  is     — y —    >  ^^^  iiot, 

dx 

as  one  might  expect     ^         . 

{dxy  : 

As  an  example  where  the  distinction  is  important,  consider 
x  =  e  —  sin  &  and  t/  =  1  —  cos  6, 
where  6  is  the  independent  variable. 


86  CALCULUS. 

B.    To  Find  the  Derivative  when  the  Function  is  Given. 

12.  Formal  work  in  diflferentiation.  The  student  should 
be  thoroughly  familiar  with  the  results  of  differentiating  all 
the  elementary  functions.  A  list  of  the  formulas  which  should 
be  memorized  is  given  below;  any  other  formulas  should  be 
worked  out  as  needed,  or  looked  up  in  a  book. 

To  establish  these  formulas,  first  prove  the  following  im- 
portant limits : 

sin  Jw      ^           J     1 .      1  —  cos  Ju 
lim  — - —  =  1,     and     lim  ^ =  0, 

provided  u  is  in  radians  ;  and 

limfl  +  iy=e  =  2.718---;* 

and  hence  prove  the  formulas  for  differentiating  the  sine  and 
the  logarithm. 

The  proofs  of  the  other  formulas  present  no  difficulty. 


*  These  lunits  having  been  established,  it  can  then  be  shown  that 

lim  sin  (u  + An) -Sinn _ ^ cos  u,  if  u  is  measured  in  degrees, 
Au^  Am  180 

=  cos  «,  if  u  is  measured  in  radians  ; 
linx  ^OS  in  +  An)- log  u  _   ^^^3^3  . . . )  1    jj  ^he  base  is  10, 

=  -,  if  the  base  is  e  =  2.718  •  •  • . 
u 

The  reason  for  choosing  the  radian  as  the  unit  angle,  and  e  as  the  base 
of  the  *'naturaP'  system  of  logarithms  is  the  simplification  in  the 
formulas  for  the  derivatives  of  the  sine  and  the  logarithm  which  results 
from  this  choice. 


CALCULUS.  87 

Rules  for   Differentiating  the  Elementary  Functions 
OF  A  Single  Variable.* 

{The  first  four  of  these  rules  are  the  fundamental  ones,  from 
which  all  the  others  can  be  derived.) 


The  differential  of  a  constant  is  zero :  — 

dk=0. 

The  differential  of  the  logarithm  to  the  base  e  of  any  function 
is  one  over  that  function,  times  the  differential  of  the  function :  — 

d{\ogex)==-^  dx  (g  =2.718. . .). 

The  differential  of  the  sine  of  any  function  (in  radians)  is 
tfle  cosine  of  that  function,  times  the  differential  of  the  function: 

d{sin  x)  =  cos  x  dx. 

The  differential  of  the  sum  [or  difference]  of  two  functions 
is  the  differential  of  the  first  plus  [or  minus]  the  differential  of 
the  second:  — 

d(u  ±v)  =  du±  dv. 


The  differential  of  a  constant  times  any  function  is  the  con- 
stant times  the  differential  of  the  function :  — 

d(Jcx)  =  k  dx.'\ 
The  differential  of  a  function  to  any  constant  power  is  the 
exponent  of  the  power,  times  the  function  to  the  power  one  less, 
times  the  differential  of  the  function :  — 

d^x^)  =  nx'^-^dx. 

if' 

Useful  special  cases  of  this  rule  are:  — 

The  differential  of  e  with  a  variable  exponent  is  e  with  the 
same  exponent,  times  the  differential  of  the  exponent :  — 

d{e^)=e'^dx  (e=2.718...). 

*  All  these  rules  remain  valid  when  the  word  ' '  derivative '  *  is  put  in 
place  of  ''differential,*'  and  the  symbol  "D"  in  place  of  " d." 

t  To  prove  this  and  the  next  five  rules,  let  y  =  the  function,  and  take 
the  logarithm  of  both  sides  before  differentiating. 


88  CALCULUS. 

The  differential  of  the  product  of  two  functions  is  the  first 
times  the  differential  of  the  second,  plus  the  second  times  the 
differential  of  the  first:  — 

d{uv)  =udv  +  V  du. 
The  differential  of  the  quotient  of  two  functions  is  the  denomi- 
nator times  the  differential  of  the  numerator,  minus  the  numer- 
ator times  the  differential  of  the  denominator,  all  divided  by 
the  denominator  squared :  — 

V  du—udv 


it)- 


v" 


The  differential  of  the  cosine  of  any  function  is  minus  the 
sine  of  that  function,  times  the  differential  of  the  function:  — 
c^(cos  x)=  —  sin  X  dx.^ 
The  differential  of  the  tangent  of  any  function  is  the  secant- 
square  of  that  function,  times  the  differential  of  the  function :  — - 
ci(tan  x)  =  sec^a;  dx.-\ 


The  differentials  of  the  inverse  sine,  the  inverse  cosine,  and 
the  inverse  tangent,  of  any  function,  are  given  by  the  following 
formulas,  which  the  student  may  put  into  words  for  himself:  — 

cZ(sin-^  x)  =  — ^^=:  dx,  J  (—  i^  ^  sin-^a:  ^  ^tt) 

A  l/l  —  X^ 

d(COS-^  iC)   = zzzrrrrrr^^  dx,  (0  ^  COS-^iT  g  Tt) 

l/l  —  X^ 

d(ta>n-^  x)  =  -— — ,  dx.  (—  i^  ^  tan-^rc  g  Iw) 

1  '\-  X 

[To  find  the  differential  of  u  to  the  i;th  power,  where  u  and  v 
are  any  functions,  let 

and  take  the  logarithm  to  base  e  of  both  sides  before  differ- 
entiating.—  Similarly,  to  find  the  differential  of  the  logarithm  of 
u  to  any  base  v,  let 

y  —  ^^Sv^}  whence  'y^  =  u; 
then  differentiate  both  sides.] 

*  Proof:  cosa;=:sin  (iTT  —  x).  f  Proof:  tan  a;  =  sin  rr/cos  a?. 

$  Proof:  Let   2/  =  sm-'a;,   that   iSj'^sin  2/  =  a;;    then   differentiate   both 
sides. — Similarly  for  the  next  two  formulas. 


CALCULUS.  89 

The  rules  cm  these  two  pages  suffice  for  the  differentiation  of 
any  elementary  function;  they  should  be  carefully  memorized. 
The  differentials  of  the  hyperbolic  functions  are  given  by 
the  following  formulas,  which  are  also  worth  remembering ; 

d  sinh  X  =  cosh  xdx;    d  cosh  x  =  sinh  x  dx ; 
d  tanh  x  =  sech2  xdx; 
hence, 

d  sinh"^  X -^  —^         ,  d  cosh~^  x=  —^^= ,  d  tanh^' 


^|x'  +  l  -Jx'-l  1-a;' 

13.  Diflferentiation  of  implicit  functions,  and  of  function! 
expressed  in  terms  of  a  parameter. 

(a)  Suppose  we  have  an  equation  connecting  x  and  y,  but 
not  giving  y  explicitly  as  a  function  of  x;  as,  for  example, 
9x2  +  42/2  =  36.  In  finding  dy/dx  in  cases  of  this  kind,  in- 
stead of  first  solving  the  equation  for  y  in  terms  of  x,  and  then 
differentiating,  it  is  usually  better  to  differentiate  the  equation 
through  as  it  stands  (remembering  that  both  x  and  y  are 
variables) ;  thus,  in  the  present  example  we  have 

18xdx-\-8ydy^0,  whence,  dy/dx = — 9x/4:y. 

This  result  can  then,  if  desired,  be  expressed  wholly  in  terms 
of  X,  by  aid  of  the  original  equation. 

(h)  Again,  suppose  y  is  given  as  a  function  of  u  and  v, 
where  u  and  v  are  both  functions  of  x;  as,  for  example, 
y  =  u^  -^v  sin  u.  Differentiating  both  sides  by  the  regular 
rules,  we  have  dy  =  2udu  +  v  cos  u  du-{-  sin  u  dv,  whence, 
collecting  the  terms  in  du  and  dv^  and  dividing  by  dx, 

dy       ,^      ,  .du    ,    .  .       .dv 

This  result  shows  how  the  rate  of  change  of  y  depends  on  the 
rates  of  change  of  u  and  v,  which  are  supposed  to  be  known. 

(c)  Finally,  both  x  and  y  may  be  given  as  functions  of  a 
third  variable,  t;  as,  x  =  F(t),  y  =  f{t).  To  every  value  of 
this  auxiliary  variable,  or  **  parameter, "  t,  there  corresponds 
a  pair  of  values  of  x  and  y,  so  that  here  again  y  is  indirectly 
determined  as  a  function  of  a;.    Of  course  if  we  can  eliminate  t 

7 


90  CALCULUS. 

we  shall  have  a  single  equation  connecting  x  and  y ;  but  it  is 
often  more  convenient  to  Iteey  the  equations  in  the  parameter 
form.  Thus,  to  find  dy/dx,  we  have  merely  to  differentiate 
both  of  the  given  equations:  dx  =  F'{t)dt,  dy  =  f'{t)dt;  and 
then  divide  the  second  result  by  the  first:  dy/dx  ^  f  (t)  /F'  (t) . 

C.     To  Find  the  Derivative  when  the  Function  Itself  is 
NOT  Given;  Setting  up  a  Differential  Equation. 

14.  In  many  cases  it  is  required  to  find  the  rate  of  change 
of  a  function  when  the  function  itself  is  not  directly  given; 
in  fact  it  is  often  easier  to  find  the  derivative  of  a  function 
than  it  is  to  find  the  function  itself. 

For  example,  a  hemispherical  bowl  of  radius  r,  full  of 
water,  is  being  emptied  through  a  hole  in  the  bottom ;  find  the 
rate  of  change  of  the  volume  of  water  drawn  off,  regarded  as  a 
function  of  the  distance,  2/,  between  the  level  of  the  water  and 
the  center  of  the  bowl.  To  compute  this  value  directly  from 
the  definition,  we  notice  first  that  the  increment  AV  produced 
in  V  by  an  increment  Ay  given  to  y  will  have  a  value  between 
7r(r2  —  y^)Ay  and  irlr^ — (2/  + Ai/)^]A2/;  dividing  either  of 
these  values  by  At/,  and  taking  the  limit  of  the  ratio  AF/Ai/, 
we  find  at  once  dV/dy  =  Tr  (r^  —  2/^),  which  gives  the  re- 
quired value  of  dV/dy  for  any  value  of  y  from  y  =  Oio  y  =  r. 

This  process  of  finding  the  derivative  directly  from  first 
principles,  as  the  limit  of  the  ratio  of  the  increments,  when 
the  function  itself  is  not  given,  is  called  **  setting  up  a  dif- 
ferential equation, '*  since  the  result  of  the  process  is  an 
equation  between  the  differentials  of  the  function  and  of  the 
argument.* 

Every  problem  of  this  kind  is  a  problem  in  finding  the 
limit  of  the  ratio  of  two  variable  quantities,  each  of  which  is 
approaching  zero;  and  in  this  connection  the  following  theo- 
rems on  infinitesimals  are  extremely  useful,  if  not  indis- 
pensable. 

*  The  problem  of  finding  the  relation  between  the  quantities  them- 
selves when  the  relation  between  their  differentials  is  known  will  be  dis- 
cussed in  the  next  chapter. 


CALCULUS.  91 

15.  Theorems  on  infinitesimals. 

Def.  Any  variable  quantity  that  approaches  0  as  a  limit  is 
called  an  infinitesimal.  For  example,  Ax,  Ay,  dx,  dy,  are 
infinitesimals. 

The  erroneous  notion  that  an  infinitesimal  is  a  constant  quantity  which 
is  ''smaller  than  any  other  quantity,  however  small,  and  yet  not  zero" 
should  be  carefully  avoided. 

Notation.  The  notation  lim  x  =  a,  or  x-^a  (read:  ^^x  ap- 
proaches a  as  a  limit  "),  means  that  x  =  a-{-e,  where  c  is  a 
variable  approaching  zero.  Thus  a  statement  expressed  in 
terms  of  '*lim"  or  **  -^  ''  can  always  be  translated  into  an 
equation^  which  can  then  be  handled  by  the  ordinary  rules  of 
algebra.  The  symbol  -»  is  preferable  to  =  and  seems  likely  to 
replace  it. 

Def.  If  a  and  ^  are  infinitesimals,  and  lim  (a/p)  =0,  then 
a  is  said  to  be  an  infinitesimal  of  higher  order  than  p. 

For  example,  if  Au=ze  -  Av,  where  e  itself  approaches  0,  then  Aw  is 
of  higher  order  than  Av.     Again,  1  —  cos  A^  is  of  higher  order  than  A^, 

If  the  difference  between  two  infinitesimals  is  of  higher 
order  than  either,  then  their  ratio  approaches  1  as  a  limit ;  and 
conversely,  if  the  ratio  of  two  infinitesimals  approaches  1^ 
then  their  difference  is  of  higher  order  than  either.  Two 
infinitesimals  having  this  relation  may  be  called  "  similar  " 
or  "  equivalent  "  infinitesimals. 

Important  examples  are  the  following:  a  convex  arc  of  a 
curve,  and  the  chord  of  that  arc,  are  '  *  similar  ' '  infinitesimals. 
Again,  sin  Ax  and  tan  Ax  are  both  ' '  similar  '  *  to  Aic,  provided 
Ax  is  in  radians. 

First  Replacement  Theorem  for  Infinitesimals.  In 
finding  the  limit  of  the  ratio  of  two  infinitesimals,  either  of 
them  may  he  replaced  hy  a  ^' similar '*  infinitesimal,  without 
affecting  the  value  of  the  limit. 

As  explained  above,  two  infinitesimals  are  "  similar  "  ;  (1) 
if  the  difference  hetween  them  is  of  higher  order  than  either; 
or  (2)  if  the  limit  of  their  ratio  is  1.  (Sometimes  the  first  test 
is  more  convenient,  sometimes  the  second.) 


92  CALCULUS. 

This  theorem  frequently  enables  us  to  replace  a  complicated 
infinitesimal,  like  7r{r-{-Ar)^Ax,  by  a  simpler  one,  as  irr^Ax; 
but  it  justifies  this  replacement  only  in  the  case  expressly 
stated  in  the  hypothesis  of  the  theorem,  namely  the  case  in 
which  we  are  finding  the  limit  of  a  ratio*  (The  fallacy  that 
*' infinitesimals  of  higher  order  can  always  be  neglected" 
should  be  carefully  guarded  against.) 

•  A  second  replacement  theorem  for  infinitesimals  will  be  given  in  the 
chapter  on  Definite  Integrals. 


CALCULUS.  ys 

D.    Applications  op  Differentiation  in  Studying  the 
Properties  op  a  Given  Function. 

16.  That  a  knowledge  of  differentiation  is  of  fundamental 
importance  in  studying  the  variation  of  a  given  function  is 
evident  from  the  following  theorems. 

Let  the  given  function  be  y==f{x). 

I.  The  value  of  the  derivative  at  any  point  shows  the  slope 
of  the  curve  at  that  point. 


Hence,  if  the  derivative  is  positive  at  any  point,  the  curve 
is  rising  at  that  point  (as  we  move  in  the  positive  direction 
along  the  axis) ;  that  is,  the  function  is  increasing.  And  if 
the  derivative  is  negative  at  any  point,  the  curve  is  falling  at 
that  point ;  that  is,  the  function  is  decreasing. 


y=/(x) 


II.  //  the  second  derivative  is  positive  at  any  point,  the 
slope  is  increasing  at  that  point,  and  hence  the  curve  is  con- 
cave upward;  and  if  the  second  derivative  is  negative  at  any 
point,  the  slope  is  decreasing  at  that  point,  and  hence  the 
curve  is  concave  downward. 


94 


CALCULUS. 


A  point  where  the  concavity  changes  sign  is  called  a  point 
of  inflexion;  at  every  such  point,  the  second  derivative  is 
0  or  00.* 

17.  Maxima  and  minima. — The  application  to  problems  in 
maxima  and  minima  is  immediate.  In  seeking  the  largest  or 
smallest  value  of  a  given  function  in  a  given  interval,  we  need 
consider  only  (1)  the  points  where  the  slope  is  zero;  (2)  the 
points  where  the  slope  is  infinite  (or  otherwise  discontinuous) ; 
and  (3)  the  end-points  of  the  interval;  for  among  these  points 
the  desired  point  will  certainly  be  found.  In  most  practical 
cases  it  will  be  a  point  where  the  slope  is  zero. 

The  conditions  of  the  problem  will  usually  show  clearly 
which  of  these  points,  if  any,  is  a  maximum  (or  a  minimum). 


18.  Multiple  roots. — The  roots,  or  the  zeros,  of  a  function, 
are  the  values  of  the  argument  for  which  the  function  becomes 


y=_f{X) 


zero.  An  inspection  of  the  figure  will  show  that  any  value  of 
X  for  which  f{x)  and  f{x)  are  both  zero  simultaneously,  will 
count  as  at  least  a  double  root. 


*  But   the  second   derivative  may  be   zero   at  points  which  are 
points  of  inflexion;  for  example,  y  =  cc*'  at  a;  =  0. 


not 


CALCULUS.  95 

19.  Small  errors. — The  following  theorem  is  very  useful  in 
discussing  the  effect,  on  a  computed  value,  of  small  errors  in 
the  data : 

III.  If  dx  is  small,  dy  and  Ay  are  nearly  equal. 

That  is,  the  difference  between  dy  and  Ay  can  be  made  as  small  as  we 
please,  in  comparison  with  dx,  by  making  dx  sufficiently  small  (except 
at  points  where  dy/dx  does  not  have  a  finite  value). 

Thus,  if  we  wish  to  find  approximately  the  error  Ay  pro- 
duced by  a  small  error  in  x,  it  will  usually  be  sufficiently 
accurate  to  compute,  instead  of  Ay,  the  simpler  value,  dy. 

In  problems  concerning  the  relative  error,  dy/y^  or  dx/x, 
it  is  often  convenient  to  take  the  logarithm  of  both  sides  of 
the  given  equation  y==f{x)  before  differentiating. 

This  class  of  problems  is  of  great  practical  value. 


CHAPTER  III. 

INTEGRATION    AS    THE    INVERSE    OF    DIFFERENTIATION.      SIMPLE 
DIFFERENTIAL  EQUATIONS. 

20.  In  many  problems  in  pure  and  applied  mathematics,  we 
have  given  the  derivative  [or  differential]  of  a  function,  and 
are  required  to  find  the  function  itself. 

Suppose  f{x)  [oT  f{x)dx]  is  the  given  derivative  [or  differ- 
ential] ;  it  is  required  to  find  a  function  F{x)  which,  when  dif- 
ferentiated, will  give  f{x)  [ov  f{x)dx].  Clearly,  if  one  such 
function  F{x)  has  been  found,  then  any  function  of  the  form 
F{x)-{-C,  where  C  is  any  constant,  will  have  the  same 
property. 

Definition. — Any  function  F{x)  whose  differential  is 
f(x)dx  is  denoted  by 

^A^)dx, 


fi 


read:  an  integral  of  f{x)dx.  The  process  of  finding  an  inte- 
gral of  a  function  is  called  integration,  or  the  inverse  of 
differentiation, 

li  F{x)  is  any  particular  integral  of  f{x)dx,  then: every 
integral  of  f{x)dx  can  be  expressed  in  the  form  F{x)  +  C, 
where  (7  is  a  constant,  called  the  constant  of  integration. 

It  can  be  shown  that  every  continuous  function  has  an  inte- 
gral; but  this  integral  may  not  (in  general,  will  not)  be  ex- 
pressible in  terms  of  the  elementary  functions.* 

Most  of  the  functions  that  occur  in  practice  can,  however, 
be  integrated  in  terms  of  elementary  functions,  by  the  aid  of 
a  table  of  integrals,  such  as  B.  0.  Peirce's  well-known  table 
of  integrals.  The  entries  in  such  a  table  can  be  verified  by 
differentiation. 

21.  Formal  work  in  integration. — The  time  devoted  to  the 
formal  work  of  integration  should  not  be  longer  than  is  nec- 

*  In  such  cases,  an  approximate  expression  for  the  integral  may  be 
obtained  by  infinite  series. 


CALCULUS.  97 

essary  to  give  the  student  a  reasonable  degree  of  expertness  in 
the  use  of  the  tables. 

The  following  integration  formulas  should  be  memorized; 
they  are  derived  immediately  from  the  corresponding  formulas 
for  differentiation. 

j  cudx  =  c  I  udx  ;      j  (w  +  v  +  •  •  •  ^dx  =  i  udx  +  j  vdx  -f  •  •  • ; 

x'^dx  =  (provided  n  4=  —  1)  ; 

(in  words:  an  integral  of  any  function  raised  to  a  constant 
power,  rfr  —  1,  times  the  differential  of  that  function,  is  equal 
to  the  function  raised  to  a  power  one  greater,  divided  by  the 
new  exponent) ; 

I  Bmxdx=—co8x;         j  cos  a; da;  =  sin  a;;         I  sec^ x dx  =  ta,n  x; 

/dx             ,                                                    r     dx  .       , 

u  =  sm  ^  a;,  or  --  cos~*a; ;  I ~  —  tan    x. 

Vl  — «'  '  '  J  1  +  a;' 

The  constant  of  integration  must  be  supplied  in  each  case. 

A  large  number  of  integrals  can  be  brought  under  the  form 
[x^dx  by  a  simple  transformation.    For  example, 

/cos^  xdx  =  /cos^  xcosxdx  =  f{l  —  sin^  x)  cos  ic  dx 
=  /cos  xdx  —  fsin^ X eos X dx  =  f cos xdx  —  /(sin  x)^d(sin  x) 

=  sin  a;  —  (sin  x)  V3. 

Similarly  for  any  odd  power  of  the  sine  or  cosine. 

The  following  integrals  are  also  important,  though  it  is  not 
worth  while  to  memorize  them  when  a  table  is  at  hand : 

j  sin'  xdx  =  ^(a;— sin  x  cos  x) ;      I  cos'  xdx  ==  ^{x  +  sin  x  cos  x) ; 

/dx        .       ,       /tt   ,  x\      ,  -       1 4-  sin  a;      r  dx       .       ^      x 

I  sinh  xdx=  cosh  x]    I  cosh  xdx  =  sinh  x;    I  sech'a;cia;=  tanha;. 


98 


CALCULUS. 


22.  Among  the  other  formulas  of  integration,  the  following 
are  perhaps  the  ones  that  occur  most  often  in  practice;  they 
are  inserted  here  for  reference,  and  especially  to  illustrate  the 
usefulness  of  the  hyperbolic  functions. 

/dx         II  ^       ,x 
-J— — ^  =  -tan  ^-, 
a^  -{-  x^       a  a' 

/dx            1   ,       a  -{-  X      1  ^      .    .  X 
-2 i  =  ^  log* =  -  tanh  ^  -, 
a^  —  x'       2a     °'  a  —  X      a  a 

/dx      __1  X  —  a  __       1  J  X 

31?  ^  a^       2a       '  X  -{-  a  a  a ' 

/dx  .   -1  a:  _,x 

■  =  sin  ^  - ,     or     =  —  cos  ^  - , 

-Sa*^  a?  a  a 

f-7==,  =  log.  (-^  H-  V^M^) ,     or     =  sinh-^  ^ , 
^    \x^  -fa*  a 

r-^=^  =  log^  {x  4-  Va:*  -  a') ,     or     =  cosh-^  ^ , 
j  Va'  —  ic*  c?a:  =  1^  he  Va*  — •  a;*  +  a'  sin~^  -    , 

or  =  -  he  V^  +  a'  +  a'  sinh~^  -    , 
r ■>!3^'c^dx  =  \[x  Va;'--a'—  a'  log^  (a;  +  Va;'— a')] , 

or  =  -  he  V^^-V—  a'  cosh"^  -    . 
2L  aj 

23.  Methods  of  Integration.  Among  the  methods  by  which 
a  given  integral  may  be  reduced  to  a  form  in  the  tables  (or 
an  integral  in  the  table  to  one  of  the  fundamental  forms),  the 
most  important  are  (1)  the  method  of  substitution  and  (2) 
the  method  of  integration  by  parts. 

In  the  method  of  substitution j  the  given  integral,  ff{x)dx, 
is  expressed  wholly  in  terms  of  some  new  variable  y  (and  dy)f 
in  the  hope  that  the  new  integral  may  be  easier  to  handle  than 
the  old  one.  The  substitutions  which  are  most  likely  to  be 
useful  are  the  following : 


CALCULUS.  99 

(a)  y  =  any  part  of  the  given  expression  whose  differential 
occurs  as  a  factor;  y  =  x^;  y  =  l/x;  y  =  smx;  y  =  Qosx; 
i/  =  tan(x/2). 

(&)  x  =  a  sin  y,  0T  =  a  tan  y,  or  =  a  sec  y,  in  expressions 
involving  ^/a^  —  ic^,  or  Va^  +  a;^,  or  y/x^  —  a^7  respectively. 

But  much  can  be  done  without  formal  substitution  of  a  new 
letter,  if  one  remembers  that  the  **  re  *'  in  the  formulas  of 
integration  may  stand  for  any  function. 

The  method  of  integration  hy  parts  is  an  application  of  the 
formula 


Cudv  =  uv —  Cvdu. 


Take  as  dv  a  part  of  the  given  expression  which  can  be 
readily  integrated ;  on  applying  the  formula,  the  new  integral 
may  be  simpler  than  the  old  one. 

The  student  should  be  practiced  in  both  of  these  methods. 

24.  Simple  differential  equations.  In  a  large  number  of 
problems  in  pure  and  applied  mathematics,  it  is  possible  to 
write  down  an  expression  involving  the  rate  of  change  of  a 
desired  function  more  readily  than  to  write  down  the  expres- 
sion for  the  function  itself.  (Compare  Chap.  II,  B.)  In 
other  words,  it  is  often  easier  to  write  down  a  relation  between 
the  differentials  of  two  variables  than  to  write  down  the  rela- 
tion between  the  variables  themselves.  Such  a  relation  con- 
necting the  differentials  of  two  or  more  quantities,  is  called  a 
differential  equation,  and  any  function  which  satisfies  the 
equation,  when  substituted  therein,  is  called  a  solution  of  the 
equation. 

Every  such  problem,  then,  breaks  up  into  two  parts:  (1) 
setting  up  the  differential  equation;  (2)  solving  that  equation. 

The  first  part  of  the  problem  has  already  been  treated  in 
Chap.  II,  B.  This  part  of  the  problem  is  too  apt  to  be  neg- 
lected in  elementary  courses;  there  is  scarcely  anything  that 
develops  real  appreciation  of  the  power  of  the  calculus  more 
effectively  than  practice  in  setting  up  for  one  *s  self  the  differ- 
ential equations  for  various  physical  phenomena. 


100  CALCULUS. 

As  to  the  second  part  of  the  problem,  namely,  the  solu- 
tion of  the  differential  equation,  the  general  plan  is  to  reduce 
the  given  equation,  by  more  or  less  ingenious  devices,  to  the 
form  dy  =  f{x)dx,  or  y  =  ff{x)dx,  and  then  to  complete  the 
solution,  if  possible,  by  the  aid  of  a  table  of  integrals.  In  a 
technical  sense,  the  differential  equation  is  said  to  be  ''solved" 
when  it  is  thus  reduced  to  a  simple  * '  quadrature ' ' ;  that  is,  to  a 
single  integration. 

The  solution  of  a  differential  equation  of  the  nth  order,  that 
is,  an  equation  involving  the  nth  derivative,  will  contain  n 
arbitrary  constants ;  to  determine  these  constants,  n  conditions 
connecting  x,  y,  y'  .  .  .,  2/^"^  must  be  known  (the  **  initial  " 
or  **  auxiliary  "  conditions  of  the  problem). 

25.  The  general  discussion  of  differential  equations  is  too 
large  and  too  difficult  a  topic  to  find  a  place  in  a  first  course 
in  the  calculus,  but  two,  at  least,  of  the  simpler  equations  are 
so  important  that  their  solution  should  be  given,  as  an  exercise 
in  integration. 

These  equations  are  the  following: 

(1)  ^  +  »'2,  =  0,  where3/'  =  |. 

The  solution  is 

2/  =  Ci  sin  {nt  +  C^)  or,  t/  =  Cg  sin  nt  +  C^  cos  n#, 

where  the  C's  are  arbitrary  constants. 

(2)  f-„',  =  0,  where  ,'  =  |. 
The  solution  is 

2/  =  Ci  sinh  {nt  +  G^),  or,  y  =  O^e"*  +  C^e""*, 

where  the  C's  are  arbitrary  constants. 

The  method  of  obtaining  these  results,  rather  than  the  re- 
sults themselves,  should  be  remembered:  namely,  multiply 
through  by  dy,  noting  that  dy/dt  ==  y\  and  integrate  each  term, 
getting  Ji/''  +  in^y^  ^  C;  then  replace  y'  by  dy/dt,  "separate 


CALCULUS.  101 

the  variables/*  and  integrate  again.  By  a  similar  method,  any 
equation  of  the  form  dy'/dt  -\-  f{y)  =  0  can  be  solved,  if  we 
can  integrate  f{y)dy. 

26.  Another  very  important   differential   equation  is  the 
equation  for  *  *  damped  vibration  ' '  : 

f +  2^1  +  ^  =  0,  where  ,'  =  ! 

The  solution  is  given  here  for  reference : 


Case  1.  If  a^  —  b^>  0,  let  m  =  V^"  —  &^  then 
y  =  Cj^e-^*  sin  (m^  +  Ca) , 
or  t/=  [Cgsin  {mt)  ^-C'^cos  {mt)]e-^*. 
Case  2.     If  a^  —  l)^  =  0, 


Case  3.     If  a^  —  h^<  0,  let  n=^/b^  —  a^;  then 

y  =  C^e-^*sinh{nt  +  C2), 

or  y  =  Cge-^^***)*  +  C^e-<»-»»>*. 

26a.  Another  important  case  is  the  linear  differential  equa- 
tion of  the  first  order : 

where  P  and  Q  are  functions  of  a;  (or  constants),  but  do  not 
contain  y.    The  solution  is  given  here  for  reference : 

2/e^  =  JQe^dx  -f  const. , 
where 

F  =  fPdx. 


CHAPTER  IV. 

INTEGRATION   AS   THE   LIMIT   OP   A   SUM.      DEFINITE   INTEGRALS. 

27.  The  limit  of  a  sum.  Many  problems  in  pure  and  ap- 
plied mathematics  can  be  brought  under  the  following  general 
form: 

Given,  a  continuous  function,  y=f(x),  from  x  =  a  to 
x  =  h.  Divide  the  interval  from  x  =  a  to  x  =  h  into  n  equal 
parts,  of  length  Ax=={b  —  a)/n*  Let  x^^x^^x^^  .  .  ,  Xn  he 
values  of  x,  one  in  each  interval;  take  the  value  of  the  func- 
tion at  each  of  these  points,  and  multiply  hy  Ax;  then  form 
the  sum: 

f(x,)AX  +  f{x,)AX+'-'+f{Xn)AX. 

Required,  the  limit  of  this  sum,  as  n  increases  indefinitely, 
and  Ax  z^  0. 

This  problem  may  be  interpreted  geometrically  as  the  prob- 
lem of  finding  the  area  under  the  curve  y  =  f{x),  between  the 
ordinates  x  =  a  and  x  =  & ;  each  term  of  the  sum  represents 


the  area  of  a  rectangle  whose  base  is  Ax  and  whose  altitude  is 
the  height  of  the  curve  at  one  of  the  points  selected.  It  is 
easily  seen  that  the  difference  between  the  sum  of  the  rec- 
tangles and  the  area  of  the  curve  is  less  than  a  rectangle 

*  It  is  not  necessary  that  the  parts  be  equal,  provided  the  largest  of 
them  approaches  zero  when  n  is  made  to  increase  indefinitely. 

102 


CALCULUS.  103 

whose  base  is  Ax  and  whose  altitude  is  constant.  This  dif- 
ference approaches  zero  as  Ax  =  0;  therefore  the  sum  of  the 
rectangles  approaches  the  area  of  the  curve  as  a  limit. 

In  this  way,  or  by  an  analytic  proof,  it  is  shown  that  the 
limit  of  the  sum  in  question  always  exists.  The  problem  then 
is,  to  find  the  value  of  this  limit. 

The  value  of  the  limit  can  always  be  obtained  by  the  fol- 
lowing fundamental  theorem,  whenever  an  integral  of  the 
given  function  f{x)  can  be  found. 

Fundamental  Theorem  of  Summation.  If  x^,  x^,  •  •  •  Xn 
are  values  of  x  ranging  from  x  =  a  to  x  =  b,  as  in  the  state- 
ment of  the  general  problem  above,  then 

lim  U(^,)Ax  +Kx,)Ax  +  . . .  4-KxjAx-]  =  F(b)  -  F(a), 

Aa:=0 

where 

F(x)^ff(x)dx 

is  any  function  whose  derivative  is  the  given  function  f{x). 

The  proof  of  this  remarkable  theorem  is  best  given  by  show- 
ing that  the  right  hand  side  of  the  equation,  as  well  as  the 
left,  is  equal  to  the  area  under  the  curve  from  x  =  ato  x  =  b; 
to  do  this,  consider  the  area  from  x  =  a  to  a  variable  point 
x=:x,  and  find  the  rate  of  change  of  this  area  regarded  as  a 
function  of  x;  hence  find  the  area  itself  as  a  function  of  re, 
determine  the  constant  of  integration  in  the  usual  way,  and 
then  put  x  =  b  in  the  result. 

Definition.  The  limit  of  a  sum  of  the  kind  described  above 
is  called  the  dej&nite  integral  of  f{x)dx  from  x  =  a  to  x  =  b, 
and  is  denoted  by 

^'"^  i:/(^.)^,    or  r^'Kx)dx. 

"=*     i=0  ^z=a 

The  function  obtained  by  the  inverse  of  differentiation  is 
called,  for  distinction,  an  indefinite  integral.  By  the  funda- 
mental theorem  just  stated,  the  definite  integral  is  equal  to  the 
difference  between  two  values  of  the  indefinite  integral : 

r^'fix^dx = r  ff(x)dx]  -  [  ff{x)dx]  . 

A/x—a  *-^  -'x=b        >-»/  -Jx=a 


104  CALCULUS. 

The  double  use  of  the  term  *' integration  *  * — meaning  in  one  case  anti- 
differentiation,  and  in  the  other  case  finding  the  limit  of  a  sum — and  the 
fundamental  theorem  connecting  these  two  distinct  concepts,  should  be 
made  thoroughly  clear.* 

The  concept  of  the  definite  integral  is  the  most  useful  con- 
cept in  the  application  of  the  calculus,  and  the  study  of 
problems  which  can  be  formulated  as  definite  integrals  may 
well  occupy  one  third  of  the  time  of  a  first  course. 

For  example,  problems  in  areas,  volumes,  surfaces,  length  of  arc, 
center  of  gravity,  moments  of  inertia,  center  of  fluid  pressure,  etc. 
Many  of  these  problems  require  two  applications  of  the  fundamental' 
theorem, 

28.  Properties  of  definite  integrals.  From  the  definition 
of  the  definite  integral  we  have  at  once : 

Ja  Jb 

rf{x)dx  +  Cf{x)dx^  rKx)dx; 

and,  by  the  aid  of  a  figure,  the  Mean  Value  theorem: 

rF(x)f(x)dx  =  F(X)  Cs(^x)dx, 

where  X  is  some  (unknown)  value  of  x  between  a  and  &,  and 
F{x)  and  f{x)  are  any  continuous  functions,  provided  f{x) 
does  not  change  sign  from  x^=^a  to  x  =  'b. 

We  have  also  the  following  important  theorem  on  change  of 
variable: 

In  evaluating  the  integral 


£ 


:6 

f(x)dx, 


if  ic  is  a  function  of  a  new  variable  ^,  we  may  replace  f{x)dx 
by  its  value  in  terms  of  t  and  dt,  and  replace  x  =  a  and  x  =  b 

*  The  use  of  the  term  in  the  sense  of  summation  was  historically  the 
earlier,  and  the  symbol  f  is  the  old  English  "long  s,"  the  first  letter 
of  *^mm," 


CALCULUS.  105 

by  the  corresponding  values  t==a  and  t  =  p,  without  altering 
the  value  of  the  integral,  provided  that  thoroughout  the  inter- 
val considered  there  is  one  and  only  one  value  of  x  for  every 
value  of  tf  and  one  and  only  one  value  of  t  for  every  value  of  x, 

29.  All  problems  leading  to  a  definite  integral  are  prob- 
lems in  finding  the  limit  of  a  sum,  each  term  of  which  is 
approaching  zero,  while  the  number  of  terms  is  increasing 
indefinitely.  "Whenever  a  function  f{x)  can  be  found,  such 
that  all  terms  of  the  sum  are  obtained  by  substituting  suc- 
cessively x^f  x^f  etc.,  in  the  expression  f{x)dx,  then  the  formu- 
lation of  the  problem  as  a  definite  integral  is  immediately 
obvious.  The  separate  terms  of  the  sum,  of  which  f{Xk)dx  is 
a  type,  are  called  elements. 

Thus,  in  finding  the  area  under  a  curve,  an  obvious  element 
of  area  is  the  rectangle  ydx;  if  the  curve  revolves  about  the 
ic-axis,  the  element  of  volume  of  the  solid  thus  generated  i? 
the  cylinder  yy^dx.  Here  y  must  be  expressed  as  a  function 
of  X  before  the  integration  can  be  completed.  Again,  in  polar 
coordinates,  the  element  of  area  is  the  sector,  ^r^ddf  where  r 
must  be  a  known  function  of  6. 

In  many  cases,  however,  the  proper  function  is  not  so 
immediately  obvious.  In  such  cases,  the  following  theorem  is 
of  great  service : 

Second  replacement  theorem  for  infinitesmals  (Theo- 
rem OF  Duhamel).  In  finding  the  limit  of  a  sum  of  positive 
terms,  each  of  which  approaches  zero  while  the  number  of 
terms  increases  indefinitely,  any  term  may  he  replaced  hy  a 
''  similar  ''  term  without  affecting  the  value  of  the  limit.  Two 
variables  a  and  p  are  called  **  similar  ''  if 

(1)  lim^=l,      or     ij(2)  lim^'-^=0. 

For  example,  let  us  find  the  weight  of  a  rod  whose  density, 

w,  and  cross-section,  A,  are  both  functions  of  x.    The  *'true 

element"  of  weight,  AW,  corresponding  to  a  given  length  ^x, 

will  certainly  lie  between  the  values  w'A'^x  and  w"A^^£^x,  where 

8 


106  CALCULUS. 

w\A'  are  the  smallest  values,  and  w'\A"  the  largest  values 
of  w  and  A  within  the  interval  from  x  =  x  to  a;  =  x  +  Ax; 
but  either  of  these  extreme  values  may  be  replaced  by  the 
simpler  value  wALx,  where  w^A  are  the  values  of  w  and  A 
at  the  beginning  of  the  interval,  for, 

w'A'Ax_        w"A"Ax  _ 
wA^x  wAAx 

Hence,  AW  itself,  which  lies  between  these  extremes,  can  be 
replaced  by  wAAx,  which  is  therefore  the  required  *' differ- 
ential element"  of  weight.*  The  total  weight  of  the  rod, 
from  a;  =  a  to  a;  =  &,  is  then  equal  to  the  definite  integral 

£=6 
wAdx; 
a 

where  w  and  A  must  of  course  be  expressed  as  functions  of  x 
before  the  integration  can  be  completed. 

In  justifying  replacements  of  this  kind  by  Duhamel's 
theorem,  sometimes  the  first  test  is  more  convenient,  some- 
times the  second.  When  once  the  common  replacements  have 
been  justified,  the  use  of  the  theorem  in  practice  rapidly 
becomes  almost  intuitive. 

30.  Approximate  methods  of  integration. — If  the  function 
f{x)  is  given  only  empirically,  the  theorem  on  evaluating  the 
definite  integral  by  purely  mathematical  means  cannot  be  ap- 
plied. In  such  cases,  an  approximate  value  of  the  definite 
integral 

may  be  found  by  plotting  the  curve  y==f{x)  on  squared 
paper,  and  estimating  the  area  by  counting  squares  (and  frac- 
tions of  squares). 

Another  method  of  approximation  is  by  Simpson's  Rule: 

•When  X  is  the  independent  variable,  it  is  immaterial  whether  we 
write  Aa?  or  dx. 


CALCULUS. 


107 


Divide  the  area  into  n  panels,  where  n  is  even,  and  number 
the  ordinates  from  1  to  w  +  1;  then,  if  Ax  is  the  width  of 
each  panel, 

Area  =  iAx  (first  ordinate  +  last  ordinate 

+  twice  the  sum  of  the  other  odd  ordinates 

+  four  times  the  sum  of  the  even  ordinates) . 

The  instrument  known  as  a  planimeter  provides  a  mechan- 
ical means  of  integration,  used  especially  in  measuring  the 
areas  of  indicator  cards. 

Another  and  very  important  method  of  approximation  is 
by  the  use  of  series;  see  the  next  chapter. 

31.  Definite  Integral  as  a  function  of  its  upper  limit. — If 
X  is  a  variable,  the  definite  integral 


represents  the  area  under  the  curve  y  =  f{x)  from  ic  =  a  to 
the  variable  ordinate  x  =  X,  and  is  therefore  a  function  of  X, 


y=/w 


say  4>{X).     By  applying  the  definition  of  derivative  to  this 
function,  it  is  easy  to  see  from  the  figure  that  </>'(X)  =/(X) : 

^jrv(-)<i.=/(X). 

Thus  <^(X)  is  one  of  the  indefinite  integrals  of  /(X). 


108  CALCULUS. 

Any  indefinite  integral  which  cannot  be  expressed  in  terms 
of  known  functions  can  always  be  written  as  a  definite 
integral  regarded  as  a  function  of  its  upper  limit,  and  its 
valu-e,  for  any  given  value  of  the  argument,  can  then  be  found 
by  one  of  the  methods  of  approximate  integration. 

The  elliptic  integrals,  the  most  important  of  which  are 

^     ==     and      I      Wl-{k')  sin' eddy 
e=o    i/l-(Jt')8in'^  J0=o     *"         ^     ^  » 

are  handled  in  this  way,  by  the  method  of  expansion  in  series. 
The  student  should  be  made  familiar  with  the  construction 
and  use  of  tables  of  the  elliptic  integrals. 

In  such  tables,  fc*  ia  usually  expressed  in  the  form  sin'  o,  which  empha- 
sizes the  fact  that  fc*  ^  1. 


CHAPTER  V. 

APPLICATIONS  TO  ALGEBRA :  EXPANSION  IN  SERIES ;  INDETER- 
MINATE FORMS. 

Note. — This  chapter  may  be  taken,  if  preferred,  immediately 
after  the  chapter  on  differentiation.  It  is  in  reality  an  exten- 
sion of  the  '* formal  work'*  of  that  chapter,  since  it  deals  with 
changes  in  the  form  of  algebraic  expressions. 

32.  Taylor's  theorem. — It  is  often  desirable  to  obtain  an 
approximate  expression  for  a  given  function,  in  the  neighbor- 
hood of  a  given  point  x==a,  in  the  form  of  a  series  arranged 
according  to  ascending  powers  of  x  —  a,  with  constant  coeffi- 
cients. For  values  of  x  near  to  a,  the  higher  powers  of  a;  —  a 
will  then  become  negligible. 

The  most  convenient  theorem  for  this  purpose  is  the  fol- 
lowing : 

Taylor's  Theorem.  If  f(x)  is  continuous,  and  has  deriva- 
tives through  the  {n-{-l)st,  in  the  neighborhood  of  a  given 
point  x  =  a,  then,  for  any  value  of  x  in  this  neighborhood, 

+     nl     ^"^      ''^    +(n  +  l)I^  ^      ' 

where  X  is  some  unknown  quantity  between  a  and  x.    The  last 
term. 


is  the  error  committed  if  we  stop  the  series  with  the  term  in 
{x  —  a)**,  and  the  formula  is  useful  only  when  this  error  be- 
comes smaller  and  smaller  as  we  increase  the  number  of 
terms. 

109 


110  CALCULUS. 

This  form  for  the  **  remainder "  B  is  easily  remembered  since  it 
differs  from  the  general  term  of  the  series  only  by  the  fact  that  the 
derivative  in  the  coeflficient  of  the  power  of  (x  —  o)  is  taken  for  x-=X 
instead  of  for  x  =  a*  (There  are  also  other  forms  of  the  remainder 
which  are  sometimes  useful.) 

33.  The  special  ease  where  a  =  0  is  called  Maclaurin's 
Theorem: 


/'w. ,  r(o)^,,     ,r(o)^„^/""'(X) 


/W=/(0)+-4f^  +  *^^'+-'-+^^a."  + 


n+l 


(n  +  1)!        ' 


where  X  is  some  unknown  quantity  between  0  and  x, 

34.  Another  special  case,  obtained  by  putting  n  =  0,  gives 

/W-/(a)=/'(Z)(x-a), 

where  again  X  is  some  unknown  quantity  between  a  and  x. 
This  theorem  is  called  the  Law  of  the  Mean,  and  is  of  great 
importance  in  the  theoretical  development  of  the  subject. 

35.  If  the  error-term  in  Taylor's  Theorem  approaches  zero 
as  n  increases,  the  formula  becomes  a  convergent  infinite  series, 
called  the  Taylor's  series  for  the  given  function,  about  the 
given  point  x  =  a. 

The  series  with  which  the  student  should  be  especially  fa- 
miliar are  the  following : 

•  The  simplest  proof  of  this  theorem  is  by  means  of  integration.  For 
example,  for  the  case  n  =  2,  we  have 


£rit)dt=r{X){x-a), 


where  X  is  some  (unknown)  constant  between  a  and  x  (as  is  evident  from 
a  figure)  j  but  also 

fj"'it)dt=f"(x)-ria), 

by  the  fundamental  theorem;  so  that 

r(x)-ria)=r(X)(x-a). 

Integrating  this  equation  twice  between  the  limits  x  =  a  and  x  =  x, 
remembering  that  f'{ci)  and  f"{X)  are  constants,  we  have  at  once: 

r^x)-r{a)-r{a){x-a)=r{X)Ux-ay, 
f{x)-f{a)-f\a){x^a)-r\a)^{x-ay=zr{X)iik{x-ay. 


CALCULUS.  Ill 


Binomial  series; 


/I    ,     Nm      1    .  .  wi(m  — 1)    ,  ,   w(m--l)(m  — 2)    ,  , 

(l  +  «)'"=l  +  wa;+  -^-j — '-x^-h  -^ ^ ^ar*  +  *--, 

provided  \x\  <  1. 
Sine  series : 

Bin  x  =  X  —  —  -\-  ^  —  Yi  "^  "  '      (a;  in  radians) . 
Cosine  series : 

rf^  iC*  X^ 

COS  a;  =  1  —  — J  +  ^  —  —  4-  •  •  •     (a;  in  radians). 
Exponential  series : 

Next  in  importance  are  the  series  for  log  (1  +  a;),  tan"*  x,  sinh  x, 
and  cosh  x. 

From  these  series  we  have  the  following  important  approxi- 
mations, when  X  is  small : 

sin  x=^  X ,     cos  x=  1 ,  etc. 

An  important  special  case  of  the  binomial  series  is  the 
geometric  series: 

—  1  -}-  X  -\-  x^  +  x^  +1"  'y    provided  \x\  <  1. 

36.  The  student  should  also  understand  the  comparison  test, 
and  the  test-ratio  test,  for  the  convergence  of  an  infinite  series, 
and  the  following  theorem  on  alternating  series :  If  the  terms 
of  a  series  are  alternatoly  positive  and  negative,  each  being 
numerically  less  than  or  equal  to  the  preceding,  and  if  the  nth 
term  approaches  zero  as  n  increases,  then  the  series  is  conver- 
gent, and  the  error  made  by  breaking  off  the  series  at  any 
given  term  does  not  exceed  numerically  the  value  of  the  last 
term  retained. 


112  CALCULUS. 

Further,  a  power  series  can  he  differentiated  or  integrated 
term  hy  term,  within  the  interval  of  convergence, 

37.  Indeterminate  forms. — The  evaluation  of  indeterminate 
forms  can  often  be  facilitated  by  the  use  of  the  following 
theorem,  in  which  f{x)  and  F{x)  are  functions  which  possess 
derivatives  at  a  given  point  x  =  a. 

Theorem  of  indeterminate  forms.  If  f(x)  and  F{x)  both 
approach  zero,  or  both  become  infinite,  when  x  approaches  a, 
then 


lim 


\.F{x)\      'iSL^^Wj 


The  second  limit  may  often  be  easier  to  evaluate  than  the  first. 
The  student  should  thoroughly  understand  the  meaning  of 

indeterminate  forms,  for  which  the  common  symbols p:,  l*,etc., 

are  merely  a  suggestive  short-hand  notation. 

Thus,  **0/0"  means  that  we  are  asked  to  find  the  limit  of 
a  function  i/  =  /(ic)/JP(ic),  when  f{x)  and  F{x)  both  approach 
zero.  Now  the  change  in  f{x)  alone  would  tend  to  decrease 
y  numerically,  while  the  change  in  F{x)  alone  would  tend  to 
increase  y ;  hence  we  cannot  tell,  without  further  investigation, 
what  the  combined  effect  of  both  changes,  taking  place  simul- 
taneously, will  be. 

Again,  the  symbol  1*  means  that  we  are  asked  to  find  the 
limit  of  a  function  y  =  f{x)^^'^\  when  f{x)  approaches  1  and 
F{x)  becomes  infinite.  Now  the  change  in  f{x)  alone  would 
tend  to  make  y  approach  1,  while  the  change  in  F{x)  alone 
would  tend  to  make  y  recede  from  1 ;  hence  we  cannot  tell,  with- 
out further  investigation,  what  the  combined  effect  will  be. 

The  student  should  thoroughly  master  in  this  way  the 
meaning  of  all  the  seven  types  of  indeterminate  forms,  namely, 

^,  —  ,   O-oo,  0^   r,    00 ",    00-00. 

U        00 

The  cases  involving  exponents  are  best  treated  by  first  find- 
ing the  limit  of  the  logarithm  of  y,  from  which  the  limit  of  y 


CALCULUS.  113 

can  then  be  obtained.    The  form  0-  oo,  or  y  =  f{x)  'F(x),  can 
be  written  as  y==^  ,y^/  ,,  or  t/=  ^  ,,\  v>  which  then  comes 

under  one  of  the  first  two  forms.     The  last  form,  oo  —  oo ,  is 
usually  best  handled  by  the  method  of  series. 

Before  applying  the  theorem  of  indeterminate  forms,  one 
should,  of  course,  try  first  to  find  the  required  limit  by  a 
simple  algebraic  transformation,  if  possible. 


CHAPTER  VI. 

APPLICATIONS  TO  GEOMETRY  AND  MECHANICS. 

In  all  applications  to  geometry,  in  which  a  curve  is  repre- 
sented by  an  equation  connecting  x  and  y,  the  scales  on  the  x 
and  y  axes  must  he  equal  (compare  §3,  footnote). 

38.  Tangent  and  normal. — The  equation  of  the  tangent  at 
any  point  (and  hence  the  equation  of  the  normal)  can  be 
written  down  at  once  when  we  know  the  slope  and  the  coordi- 
nates of  the  point  of  contact. 

Again,  to  find  the  suhtangent  or  subnormal  at  any  point, 
we  have  simply  to  find  the  ordinate  and  the  slope  at  that  point, 
and  then  solve  a  right  triangle. 

39.  Differential  of  arc.  If  5  =  length  of  arc  of  the  curve 
y=zf(x)y  measured  from  some  fixed  point  A  of  the  curve, 
then  s,  like  y,  is  a  function  of  x,  and  we  may  ask  what  is  the 


x,i-Zix 


rate  of  change  of  s  with  respect  to  x,  that  is,  what  is  the  value 
of  ds/dx.  Now  ds/dx  =  \\m  (As/Ax),  and  in  finding  this 
limit  we  may  replace  the  arc  As  by  its  chord,  V  (^^)^+(^2/)* ; 
hence  ds/dx  =  lim  V1+  {Ay /Ax) 2  =  y  1  -J-  (dy/dx) *,  or 


ds  =  V{dxy+{dyy, 
114 


CALCULUS.  115 

as  indicated  in  the  figure.    This  formula,  and  the  correspond- 
ing relations 

dx  =  ds  cos  <^,      dy  =  ds  sin  <^, 

are  important,  and  are  readily  recalled  to  mind  by  the  figure. 
In  the  case  of  a  circle  of  radius  r,  if  dd  =  the  angle  at  the 
center,  subtended  by  the  arc  ds,  then 

ds  =  rdd, 

provided  the  angle  is  measured  in  radians. 

40.  Again,  in  case  of  a  curve  whose  equation  is  given  in 
polar  coordinates,  r  =  f{0),  we  see  at  once  from  the  figure,  by 
the  aid  of  the  replacement  theorem,  that 

ds  =  ^{dry  4-  (rdoy     and     tan  i/r==  '^~, 

where  if/  is  the  angle  which  the  tangent  makes  with  the  radius 
vector  produced. 


41.  Radius  of  Curvature. — Consider  the  normal  to  a  given 
curve  at  a  given  point,  P,  and  also  the  normal  at  a  neighbor- 
ing point,  Q,  These  two  normals  will  intersect  at  some  point 
C  on  the  concave  side  of  the  curve ;  and  as  Q  approaches  P, 
along  the  curve,  this  point  C  will  (in  general)  approach  a 
definite  position  0  as  a  limit.  The  circle  described  with  a 
center  at  this  point  C  and  radius  equal  to  CP  will  fit  the 
given  curve  more  closely,  in  the  neighborhood  of  the  point  P, 
than  does  any  other  circle.  This  circle  is  called  the  osculating 
circle,  or  the  circle  of  curvature,  at  the  point  P ;  its  center  G 
is  called  the  center  of  curvature^  and  its  radius  CP  is  called 
the  radius  of  curvature,  at  the  point  P. 


116  CALCULUS. 

The  radius  of  curvature  may  thus  be  taken  as  a  measure  of 
the  flatness  or  sharpness  of  the  curve ;  the  smaller  the  radius 
of  curvature,  the  sharper  the  curve. 

The  length  of  the  radius  of  curvature,  B,  at  any  point  P  is 
most  readily  found  as  follows :  In  the  triangle  PC'Q,  we  have 
C'P/PQ  =  8111  Q/sin  A<^,  where  A<f>  is  the  angle  between  the 
normals  (or  between  the  tangents)  at  P  and  Q.  Therefore 
iiJ  =  lim  CT  =  lim  (chord  PQ /sin  A<f>)  sin  Q;  or,  replacing 


the  chord  by  the  arc  As,  and  sin  A(f>  by  A<^,  and  noticing  that 
Q  is  approaching  90°,  so  that  lim  sin  0  =  1,  we  have 
ii;  =  lim  (As/A</)),  or, 

This  important  formula  is  readily  recalled  to  mind  from  the 
figure,  if  one  thinks  of  the  arc  As  as  approximately  a  circular 
arc. 

To  express  B  in  terms  of  x  and  y,  we  have  only  to  remember  that 
ds=Vidxy  +  {dyy=i  \/l  +  y'^dx,  and  tan  <t>  =  dy/dx^='g',  whence 
d<t>  =  y"dx/{l-\-y'^);  then 

^^(1+2/^ 

y" 


CALCULUS.  117 

Def.  The  curvature  of  a  curve  at  a  point  is  defined  as  the 
rate  at  which  the  angle  <f>  is  changing  with  respect  to  the 
length  of  arc  s ;  that  is, 

curvature  =  -^  =  ^« . 
as       B 

If  the  slope  of  the  curve  is  small,  the  curvature  is  approxi- 
mately equal  to  3/". 

Def.  The  locus  of  the  center  of  curvature  is  called  the 
evolute  of  the  curve. 

The  normals  to  the  given  curve  are  tangent  to  the  evolute, 
and  the  given  curve  may  be  traced  by  unwinding  a  string  from 
the  evolute. 

42.  Velocity  and  acceleration. — Consider  a  particle  moving 
along  a  straight  line.  Its  distance  from  the  origin  is  a  func- 
tion of  the  time : 

x  =  F{t), 

The  velocity  of  the  particle  is  the  rate  of  change  of  its 
distance : 

y  =  dx/dt  =  F'{t)=:x'. 

The  velocity  will  be  positive  or  negative,  according  as  the 
particle  is  moving  forward  or  backward  along  the  line. 

The  acceleration  of  the  particle  is  the  rate  of  change  of  its 
velocity: 

A  =  dv/dt  =  F"{t)=x". 

The  acceleration  will  be  positive  or  negative  according  as 
the  velocity  is  increasing  or  decreasing  (algebraically). 

If  a  particle  is  moving  along  a  plane  curve,  we  must 
consider  the  components  of  its  motion  along  two  fixed  axes. 
The  components  of  acceleration  along  the  x-  and  2/-axes 
are  x"  and  y" ;  the  components  of  acceleration  along  the 
tangent  and  normal  are  dv/dt  and  v^/B,  respectively,  where 
V  =  yx'^  -\-  y'^  =  the  path  velocity,  and  B  =  the  radius  of 
curvature. 

It  should  be  carefully  noticed  that  dv/dt  is  not  the  whole  acceleration, 
but  only  that  component  of  the  acceleration  which  lies  along  the  tangent. 


118  CALCULUS. 

The  importance  of  this  application  in  problems  in  mechanics 
is  obvious. 

Note. — As  explained  in  the  preface  of  this  report,  these 
pages  are  intended  merely  to  give  a  resume  of  the  working 
principles  of  the  calculus  with  which  the  student  should  be 
perfectly  familiar  after  having  taken  a  course  in  this  subject. 
The  main  part  of  the  work  of  such  a  course  should  he  prob- 
lems done  hy  the  students — each  problem  being  solved  on  the 
basis  of  the  small  number  of  fundamental  theorems  here 
mentioned. 


discussion  on  mathematics  keport.  119 

Discussion. 

Professor  Chas.  0.  Gunther:  It  seems  to  me  that  in  this 
report  some  mention  should  be  made  of  imaginary  and  com- 
plex quantities.  A  little  knowledge  of  these  quantities  can, 
for  instance,  be  utilized  to  good  advantage  by  applying  it  to 
that  part  of  the  calculus  known  as  integration.  In  fact,  in- 
tegration can  be  simplified  to  the  extent  of  eliminating  the 
usual  * '  reduction  formula  ' '  and  rendering  the  use  of  tables 
of  integrals  unnecessary. 

As  found  in  text-books  in  general,  there  are  three  cases  for 
which  the  expression 

dy  =  cos^e  ^nJ'ede  (1) 

can  be  easily  integrated.  Two  of  these  cases  include  frac- 
tional values  for  h  and  k.  All  other  cases  in  which  h  and  k 
are  integers  can  either  directly,  or  by  means  of  a  single 
imaginary  trigonometric  substitution  (tan  ^  =  i  sin  a,  in  which 
a  is  an  imaginary  quantity),  be  reduced  to  one  or  more  of  the 
three  cases  just  referred  to. 

The  general  binomial  differential  expression 

dy  =  x'^{a  +  'bx'*)P^^dx  (2) 


is  only  another  form  of  (1)  since  V^  +  ^^"  can  always  be 
represented  by  one  of  the  three  sides  of  a  right  triangle  and 
therefore  expressed  as  a  trigonometric  function  of  one  of  the 
acute  angles  of  the  triangle. 

To  make  this  transformation  the  student  must  know  the 
relation  between  the  hypotenuse  and  the  two  sides  of  a  right 
triangle,  the  values  of  the  trigonometric  functions  of  an  angle 
in  terms  of  the  sides  of  a  right  triangle,  and  the  rules  for 
differentiation. 

Differential  expressions  involving  trinomial  surds  may  be 
rationalized  in  a  similar  manner. 

The  expressions 

^=6- cos  6a;,  (3) 

^=6- sin  6a;,  (4) 


120  DISCUSSION  ON  MATHEMATICS  REPOKT. 

may  be  integrated  with  great  facility  if  complex  quantities 
are  employed,  because  e'^'^ coshx  and  e'^^sinhx  are  the  rec- 
tangular components  of  a  vector  whose  modulus  is  e"*  and 
whose  argument  is  hx.  The  integrals  of  (3)  and  (4)  are  found 
from  the  integral  of 

in  which  s  is  a  complex  variable  of  the  form  y  +  iy.  The 
integral  of  (5)  is  readily  found  to  be 

Aa+ib)x 

in  which  Cn-u  •  •  •  Cj,  Cq,  are  constants  of  the  form  6  =  0  +  iC. 
Equation  (6)  may  be  written 

^ = ^a'^b-'r  '*""" '""'  *'  +  ^"-'''"''  +  •  •  •  +  ^.^ + c;-  (7) 

The  integral  of  (3)  is  the  real  part  of  (7)  and  the  integral  of 
(4)  is  the  imaginary  part  of  (7)  divided  by  i. 
Again  in  differential  equations  we  find  the  linear  equations 

dv 

—  -\.  ay  —  b  cos  nx,  (8) 

dy  ,  . 

—  -f  ai/  =  6  sm  nxj  (9) 

and  their  solutions  can  be  obtained  from  the  solution  of  the 
equation 

-^  +  az  =  be'^,  (10) 

in  which  z  =  y  -{-  iy. 

The  foregoing  illustrates  a  few  of  the  applications  of  com- 
plex and  imaginary  quantities,  and  includes  a  first  treatment 
of  hyperbolic  functions  as  trigonometric  functions  of  imagi- 
nary quantities. 

Some  little  consideration  should  also  be  given  to  the  com- 
plex and  imaginary  branches  of  certain  curves,  as  for  example. 


DISCUSSION  ON  MATHEMATICS  REPORT.  121 

the  circle,  the  ellipse,  and  the  hyperbola.  It  should  be  noted 
that  the  equation  of  the  circle  x^  -\-y^  =  a^  is  also  the  equation 
of  an  imaginary  hyperbola  for  values  of  a;  >  a  and  <  —  a. 
This  is  important,  since  of  the  three  forms  of  binomial  surds 
V«^  —  ic^  Va^-\-x^,  V^  —  <*^)  the  first  is  obtained  from  the 
equation  of  the  circle  x^  -\-y^  =  a^,  and  the  latter  two  from 
the  equation  of  the  hyperbola  x^  —  y^  =  a^-^  but  all  three  are 
obtained  from  the  equation  of  the  circle  if  imaginary  quanti- 
ties are  made  use  of. 

Professor  J.  E.  Boyd :  I  want  to  emphasize  everything  Pro- 
fessor Gunther  just  said  about  the  use  of  complex  quantities. 
We  cannot  derive  a  formula  for  an  eccentrically  loaded  long 
column  without  the  use  of  them  j  we  cannot  make  alternating 
current  calculations  without  them.  A  student  might  as  well 
learn  how  to  use  them.  I  endorse  what  he  says  about  the  use  of 
integral  tables  in  teaching  calculus.  Our  professors  in  calcu- 
lus last  year  adopted  a  book  that  advised  the  use  of  tables. 
This  year  a  book  of  the  other  type  was  selected.  We  did  not 
use  the  tables  any  more  than  was  absolutely  necessary  and 
found  the  result  satisfactory.  The  student  does  not  need 
tables  often,  except  to  make  use  of  the  several  transformations. 

Professor  F.  L.  Emory:  The  average  student  is  vastly  lack- 
ing in  a  knowledge  of  the  use  of  logarithms.  He  also  lacks 
the  ability  to  read  trigonometric  formulaa  from  the  triangle. 

The  tendency  of  the  report  is  to  include  more  material  than 
can  be  covered  in  an  engineering  course.  I  would  be  satisfied 
to  have  a  little  more  training  in  a  few  principles  which  stu- 
dents must  know  so  well  that  they  have  confidence  in  their 
knowledge.  One  of  the  most  serious  difficulties  that  I  encoun- 
ter is  with  the  constant  of  integration.  This  is  largely  the 
fault  of  the  text-books.  I  have  a  grievance  against  the  text- 
book writer  who  omits  the  constant  in  all  cases,  supplemented 
by  the  remark  that  it  should  always  be  added.  We  cannot 
expect  the  student  to  remember  a  footnote  to  be  applied  with 
each  operation. 

Professor  J.  B.  Webb:  I  am  pleased  to  hear  what  Pro- 
fessor Emory  said  about  the  constant  of  integration  and  his 
9 


122  DISCUSSION  ON  MATHEMATICS  KEPORT. 

explanation  of  the  difficulty,  but  I  think  the  trouble  is  more 
in  the  teaching  than  in  the  text-books.  In  reading  Mr.  F.  W. 
Taylor's  book  on  his  system,  I  was  interested  in  one  of  the 
illustrations  which  he  uses.  He  takes  the  case  of  loading  cars 
with  pig  iron,  where,  by  the  application  of  his  system  he 
about  tripled  the  amount  that  a  man  could  do  in  a  day,  and  at 
the  same  time  enabled  the  man  to  earn  more  money.  One  of 
the  first  things  he  did  was  to  examine  the  men  that  were  in  the 
gang,  and  he  found  that  but  one  man  in  eight  was  suitable 
for  this  work.  He  used  only  those  who  were  fitted  for  it.  We 
have  about  the  same  proportion,  perhaps,  of  the  unfit  in  our 
classes,  and  the  ones  fitted  for  engineering  could  do  three 
times  the  work  and  do  it  better  if  our  classes  were  conducted 
on  the  Taylor  system. 

I  have  had  some  interesting  experiences  with  the  complex 
variable.  Having  studied  the  subject  in  Germany  in  1878- 
1880,  on  my  return  to  this  country  I  tried  to  teach  its  use. 
Objections  were  made  by  those  not  acquainted  with  the  sub- 
ject, that  it  was  too  advanced  and  of  little  practical  use,  so 
that  it  proved  to  be  harder  to  convince  the  average  American 
teacher  of  its  importance  than  to  arouse  the  interest  of  intelli- 
gent students.  Some  of  the  professors  were  convinced,  but 
that  was  where  the  trouble  lay.  If  a  student  was  conditioned 
because  he  did  not  get  through  with  his  mathematics,  some 
said  I  taught  '  *  over  his  head  ' '  and  gradually  the  standard 
would  be  forced  down.  The  trouble  with  the  present  schools 
is  that  they  want  too  many  students  and  are  going  to  hold 
all  they  have  and  get  more  if  they  can.  They  do  not  call  out 
the  seven  and  keep  the  one.  After  Dr.  Steinmetz,  a  layman, 
produced  his  book  on  the  treatment  of  alternating  currents, 
using  complex  variables,  there  was  less  objection  made  to 
them.  Now  I  say  it  is  a  disgrace  that  it  should  be  necessary 
for  a  layman  to  show  professional  teachers  that  a  certain  part 
of  mathematics  is  needed.  What  we  should  do  is  to  eliminate 
the  students  who  are  not  capable  of  profiting  by  what  we  know 
should  be  taught,  and  then  hold  the  others  to  a  high  standard. 

We  expect  too  much  of  the  student  who  takes  calculus.    Of 


DISCUSSION  ON  MATHEMATICS  EEPOKT.  123 

a  semester  in  calculus  at  least  one-half  is  spent  in  reviewing 
previous  mathematics.  A  course  in  calculus  is  an  excellent 
review  of  geometry,  algebra  and  especially  of  trigonometry, 
and  at  its  close  we  should  not  expect  the  average  student  to 
know  much  more  about  it  than  he  did  about  trigonometry  at 
the  start. 

This  committee  was  appointed  to  see  what  was  the  matter 
with  the  teaching  of  mathematics.  They  imply  that  good  text- 
books are  lacking.  I  cannot  agree  with  this  and  would  rather 
have  one  of  the  old-fashioned  text-books  than  those  outlined 
in  their  report.  If  they  intend  to  give  simply  a  list  of  subjects 
that  students  should  be  drilled  in,  well  and  good;  but  if  the 
report  intends  to  prescribe  the  methods  of  thought  and  of 
logical  deduction,  to  be  used  in  those  subjects,  then  I  think  it 
is  all  wrong. 

Professor  Magruder:  The  introduction  to  the  report  states 
clearly  the  purpose  of  the  syllabus. 

Professor  W.  J.  Risley:  The  suggestions  that  have  been 
made  here  this  afternoon  are  very  good.  I  am  in  favor  of  a 
section  on  imaginary  quantities.  When  I  approached  some  of 
the  Harvard  professors  of  engineering  subjects  I  found  that 
they  wanted  their  students  to  perform  vector  addition  analyt- 
ically. They  said  that  the  teachers  of  mathematics  werft 
teaching  a  lot  of  things  of  which  little  or  no  use  was  made 
later.  To  a  great  extent  Professor  Webb  was  right  in  stating 
that  he  had  to  teach  the  professors  of  engineering  what  they 
ought  to  teach,  in  order  that  they  might  understand  some  of 
the  mathematics  which  he  attempted  to  send  to  them.  On  the 
other  hand,  sometimes  the  professors  of  engineering  have  to 
teach  the  professors  of  mathematics  some  things  that  they 
don't  know  that  their  students  ought  to  know.  Neither  set  is 
to  be  criticized  too  severely  unless  they  are  unwilling  to  learn 
when  the  right  way  is  pointed  out. 

Principal  Arthur  L.  Williston:  I  was  very  much  inter- 
ested in  what  Professor  Webb  said  a  moment  ago,  referring 
to  Mr.  Taylor's  work  and  his  method  of  culling  out  one  man 


124  DISCUSSION  ON  MATHEMATICS  REPORT. 

of  the  eight  who  was  especially  adapted  for  a  particular  kind 
of  work,  using  him  intensively  on  that  kind  of  work,  and  find- 
ing tasks  for  which  the  other  seven  are  fitted.  That  idea  is 
really  at  the  bottom  of  all  of  our  difficulty  in  this  discussion 
of  teaching  mathematics  to  engineers,  which  we  have  had 
almost  since  we  began  trying  to  teach  engineers.  As  there  are 
few  men  of  the  naturally  analytical  kind  that  Professor  Webb 
describes  it  does  not  make  much  difference  what  sort  of 
methods  we  use  with  them.  As  a  matter  of  fact  a  very  small 
proportion  of  the  men  who  form  the  body  of  eminent  engi- 
neers have  that  type  of  mind.  We  all  know  the  sort  of  fellow 
who  thrives  on  complex  quantities.  And  I  am  sure  the 
majority  of  those  here  will  bear  me  out  in  my  statement  that 
a  very  small  proportion  of  the  successful,  eminent  engineers 
of  this  country  are  of  that  kind.  The  ideal  plan  would  be  to 
separate  those  fellows  from  the  mass  and  give  them  a  course 
in  real  mathematics.  They  would  like  it  and  it  would  be  a 
pleasure  to  the  instructors  to  teach  them.  But  let  us  take  the 
other  group.  For  the  most  part,  the  man  who  is  going  to  be  a 
successful  engineer  in  industrial  work  is  a  practical,  concrete 
man.  He  does  not  handle  imaginary,  complex,  abstract  quan- 
tities easily.  And  yet  that  is  the  very  type  of  mind  that  the 
world  wants  in  its  important  industrial  activities.  Those  fel- 
lows, who,  by  the  way,  constitute  the  great  majority,  want 
mathematics  not  as  an  analytical  light  but  simply  as  a  neces- 
sary evil,  if  you  please,  as  a  tool  that  they  must  use.  If  in 
our  talking  and  our  thinking  we  could  learn  to  talk  of  mathe- 
matics as  two  subjects,  one  thing  for  the  first  type,  another 
for  the  second,  it  would  simplify  all  our  discussion.  It  is 
absolutely  futile  to  attempt  to  teach  the  first  kind  of  mathe- 
matics to  three  out  of  four  young  men  who  will  be  good  engi- 
neers whether  the  colleges  turn  them  out  as  fitted  to  be  engi- 
neers or  not.  They  are  going  to  be  engineers.  As  I  under- 
stand it,  the  work  of  this  committee  has  been  to  some  extent 
a  movement  toward  trying  to  get  the  teaching  of  mathematics 
for  engineers  differentiated  from  the  teaching  of  pure  mathe- 
matics.   I  am  sorry  that  the  difference  is  not  more  marked. 


DISCUSSION  ON  MATHEMATICS  BEPOET.  125 

Professor  E.  R.  Maurer:  I  prefer  to  hear  a  teacher  of 
mathematics  discuss  this  syllabus,  because  he  can  see  it  in  the 
light  of  his  experience  in  teaching  the  subject.  To  be  sure, 
others  have  good  ideas  as  to  what  knowledge  and  training 
engineers  ought  to  have  in  mathematics,  but  they  fail  to  ap- 
preciate the  difficulties  of  teaching  the  subject.  So,  between 
two  criticisms,  one  offered  by  teachers  of  mathematics  and  the 
other  by  teachers  who  have  never  taught  mathematics,  I  place 
more  confidence  in  the  former.  In  estimating  the  value  of 
mathematical  instruction  we  are  apt  to  forget  that,  in  many 
schools,  particularly  the  large  ones,  more  or  less  inexperienced 
men  are  employed  as  instructors  in  the  departments  of  mathe- 
matics. The  results  suffer  on  that  account.  In  addition  there 
is  the  poor  quality  of  the  working  material.  I  try  to  be  chari- 
table when  I  judge  the  students  that  come  to  me  from  the  de- 
partment of  mathematics  on  those  two  accounts.  Many  of  the 
boys  have  had  their  training  at  the  hands  of  inexperienced 
men  and  many  have  very  little  mathematical  talent.  I  think 
the  syllabus  is  good  as  a  list  of  topics  with  which  all  engineer- 
ing students  ought  to  be  familiar.  I  agree  with  Professor 
Webb  in  that  we  ought  not  to  set  this  up  as  a  subject  matter 
for  all  teachers  of  mathematics  to  use  and  not  depart  from  it 
in  any  particular.  The  teacher  of  mathematics,  or  of  any  sub- 
ject in  an  engineering  school  ought  to  understand  his  subjects 
well  enough  to  get  up  his  own  syllabus,  if  necessary. 

The  President:  An  informal  committee  of  instructors  in 
the  University  of  Illinois,  formed  of  a  dozen  men  representing 
the  department  of  mathematics,  mechanics,  civil  engineering, 
electrical  engineering  and  mechanical  engineering,  made  a 
careful  study  of  the  report  of  the  Mathematics  Committee  to 
see  whether  the  syllabi  covered  the  ground  which  these  pro- 
fessors thought  should  be  covered  in  class.  In  general  I  may 
say  that  they  approve  almost  wholly  of  the  contents  and  in 
general  of  the  matters  of  emphasis  as  to  what  part  should  be 
well  understood,  what  other,  only  partly  known.  With  your 
permission  I  shall  include  this  report  in  the  discussion. 

The  committee  of  University  of  Illinois  instructors  selected 


126  DISCUSSION  ON  MATHEMATICS  KEPORT. 

to  discuss  the  preliminary  report  of  the  Committee  on  the 
Teaching  of  Mathematics  to  Students  of  Engineering  submit 
the  following  recommendations: 

Since  the  syllabi  are  meant  to  embody  the  minimum  equip- 
ment in  mathematics  of  a  good  engineer,  they  have  been  dis- 
cussed from  that  point  of  view.  But  it  is  the  opinion  of  the 
committee  that  much  could  be  gained  by  publishing  a  list  of 
topics  that  should  be  included  in  the  courses  discussed,  and 
emphasizing  by  a  star  those  which  are  *  *  so  essential  that  every 
engineering  student  should  have  them  so  firmly  fixed  in  his 
memory  that  he  will  never  need  to  look  them  up  in  a  book." 
The  discussions  of  the  committee  were  confined  to  the  syllabi 
which  are  printed  in  the  Proceedings,  i.  e.,  Algebra,  Trigo- 
nometry, Analytic  Geometry,  and  Calculus.  Section  numbers 
refer  to  the  sections  as  published  in  the  syllabus. 

Algebra. 

1.  Under  factoring  some  mention  should  be  made  of  the 
important  cases  of  collecting  coefficients,  and  of  quadratic 
trinomials. 

2.  Important  principles  and  rules  should  be  given  in  trans- 
lated word  form  as  well  as  in  symbolic  form  (as  is  done  once  on 
page  8  and  in  the  differentiation  rules  in  the  calculus  syl- 
labus). Students  often  fail  to  get  the  full  meaning  of  sym- 
bolic forms.  The  operations  with  fractions  and  the  definitions 
and  laws  of  exponents  especially  need  statement  in  word  form. 

3.  If  algebra  follows  trigonometry,  the  three  forms  for 
imaginaries  should  be  included. 

4.  The  notions  equality,  identity  and  equation  should  be 
carefully  differentiated. 

5.  The  principles  of  equivalent  equations  should  be  in- 
cluded, for  a  student  should  know  what  operations  introduce 
or  take  out  roots. 

6.  More  emphasis  is  needed  on  the  ** completing  the  square'* 
process,  for  it  is  often  needed  later  in  integration  and  analytics 
when  no  solution  is  required. 

7.  Harmonic  progression  should  be  omitted. 


DISCUSSION  ON  MATHEMATICS  KEPORT.  127 

Trigonometry, 

1.  The  committee  agrees  that  the  syllabus  is  satisfactory 
and  probably  is  complete  enough  for  the  average  engineer. 
Some  members  expressed  a  desire  for  the  memorization  of 
more  formulas  as  particularly  useful  to  electric  engineers. 

2.  Some  members  desired  greater  stress  on  the  visible  hand- 
ling of  formulae.  By  visible  is  meant  graphical  so  far  as  the 
expression  of  relationship  and  formulae  can  be.  For  ex- 
ample the  student  should  not  so  much  remember  the  six  fun- 
damental definitions  as  formulae  as  he  should  remember  the 
defining  triangle  and  its  ratios.  The  same  idea  should  be 
carried  throughout. 

Analytic  Geometry. 

1.  The  syllabus  states  in  the  introduction,  ''This  syllabus 
is  confined  mainly  to  the  conic  sections;  but  a  satisfactory 
course  in  analytic  geometry  should  include  also  the  study  of 
many  other  curves.''  This  committee  believes  that  the  syl- 
labus would  be  improved  by  including  the  most  important  of 
these  ''many  other  curves"  including  the  sorcalled  engineer- 
ing curves. 

2.  The  equation  of  a  straight  line  passing  through  two  given 
points  should  be  included. 

3.  The  equation  of  a  straight  line  should  be  written  in  such 
a  form  and  taught  in  such  a  manner  that  all  constants  of  the 
line  are  readily  determined. 

4.  The  method  of  treating  the  conic  sections  in  the  syllabus 
is  commended.  For  obtaining  a  proper  facility  in  handling 
the  practical  applications  of  these  curves,  it  is  desirable  to 
study  each  form  separately  even  at  the  expense  of  the  addi- 
tional time  that  is  required  when  this  method  is  employed. 
The  properties  of  these  curves  as  given  are  amply  sufficient. 

5.  The  geometrical  construction  of  the  conies  should  be  in- 
cluded and  given  more  than  a  mere  reference. 

6.  In  the  transformation  of  coordinates  the  method  rather 
than  the  equations  should  be  remembered. 

7.  The  subject  matter  in  articles  46-54  is  not  that  which  a 


128  DISCUSSION  ON  MATHEMATICS  REPORT. 

student  should  remember,  but  belongs  to  that  class  of  things 
which  can  easily  be  referred  to  when  required. 

8.  Much  greater  emphasis  should  be  placed  upon  work  in 
polar  coordinates. 

9.  It  is  desirable  for  the  student  to  be  familiar  with  cylin- 
drical coordinates  and  the  committee  commends  the  inclusion 
of  these  coordinates  in  the  syllabus. 

10.  Great  stress  should  be  laid  upon  representation  with 
space  coordinates.  Any  single  equation  in  space  coordinates 
represents  some  surface.  If  the  equation  is  in  three  variables 
the  surface  may  be  any  form,  if  in  two  variables  the  surface  is 
a  cylinder,  if  in  one  variable  the  surface  is  a  plane  or  a  system 
of  planes  parallel  to  one  of  the  coordinate  planes.  Great 
emphasis  should  be  placed  upon  the  fact  that  it  requires  a 
pair  of  simultaneous  equations  to  determine  a  line  in  space. 

11.  Article  71  should  be  omitted  from  the  syllabus,  though 
included  in  a  course  in  Analytic  Geometry. 

12.  In  the  first  sentence  of  the  second  paragraph  of  the 
introduction  the  phrase  **a  course  should  consist  chiefly  of 
problems"  should  be  changed  to  read  *'a  large  number  of 
problems  should  supplement  the  treatment  of  general  prin- 
ciples. ' ' 

Calculus. 

The  committee  reports  very  favorably  on  the  syllabus  for 
the  first  part  of  calculus.  A  subcommittee  drew  up  a  synop- 
sis of  a  course  in  calculus  before  reading  the  syllabus  as 
printed  in  the  Bulletin.  The  two  did  not  differ  in  many 
essential  details.  The  main  question  that  came  up  was  whether 
a  topic  was  included  under  ''those  facts  and  methods  which 
every  student  should  have  so  firmly  fixed  in  his  memory  that 
he  will  never  need  to  look  them  up  in  a  book, ' '  or  simply  under 
"those  topics  included  in  an  elementary  course  in  calculus.'* 
These  two  classes  are  referred  to  below  as  first  and  second 
classes.  The  specific  changes  suggested  in  the  syllabus  are  as 
follows ; 

1.  Section  5.  Hyperbolic  functions  should  be  included  in 
the  second  of  the  above  classes.    Mnemonic  rules  for  changing 


DISCUSSION  ON  MATHEMATICS  REPORT.  129 

a  trigonometric   formula   to   the   corresponding   formula  in 
hyperbolic  functions  should  be  included. 

2.  Use  arc  sin  x,  arc  cos  x,  etc.,  instead  of  sin-^  x,  cos-\  etc. 

3.  Section  21  (Formal  work  in  integration).  Tables  of 
integrals  should  not  be  used  until  the  student  has  had  con- 
siderable practice  in  formal  integration. 

4.  Include  in  section  22,  integration  by  separation  into 
partial  fractions. 

5.  Much  practice  in  differentiation  and  integration  with 
respect  to  variables  represented  by  symbols  other  than  x,  y,  z 
should  be  given. 

6.  In  connection  with  differential  equations  (Sections  24, 
25,  26)  use  d^y/dx^  instead  of  dy'/dx. 

7.  Include  linear  differential  equations  of  first  order  in  con- 
nection with  sections  25,  26. 

8.  Include  sections  15  (Theorems  on  infinitesimals),  22 
(Integration  formulas),  35  (Theorem  of  Duhamel),  38  (Sub- 
tangents,  subnormals,  etc.),  41  (Curvature),  in  the  second  of 
the  above  classes. 

9.  Include  angular  velocity  and  acceleration  in  section  42. 

10.  We  particularly  commend  sections  7,  12  (note)  and  14. 

Professor  A.  M.  Buck:  A  good  many  people,  and  espe- 
cially some  who  are  mathematicians,  forget  that  with  the  engi- 
neering student  mathematics  is  a  subject  that  is  taken  not  for 
its  own  sake,  but  in  order  that  problems  can  be  solved  after- 
wards. If  we  take  the  view-point  of  the  students  we  find  that 
they  appreciate  this  point  better  than  their  teachers  do.  Stu- 
dents have  told  me  that  they  could  not  get  along  in  mathe- 
matics because  they  did  not  know  what  use  they  were  going  to 
make  of  it.  Had  it  been  brought  to  their  attention  that  the 
mathematics  would  have  some  application  to  their  engineering 
work  they  would  have  gone  into  it  with  good  spirit  and  would 
have  obtained  more  benefit  from  the  work.  Taking  it  as  an 
abstract  study  they  simply  would  not  give  it  the  necessary 
time.  If  the  teacher  of  mathematics  will  look  at  his  subject 
from  an  engineering  view-point  and  see  that  those  things 
which  he  teaches  are  to  be  used  as  tools  and  that  the  better  the 


130  DISCUSSION  ON  MATHEMATICS  BEPOET. 

student  has  his  tools  in  hand  the  better  work  he  can  do,  then 
there  will  be  an  improvement  in  the  teaching. 

Professor  H.  R.  Thayer:  I  am  going  to  state  my  opinion 
from  the  view-point  of  the  engineer.  I  have  spent  more  time 
outside  in  practice  than  I  have  in  teaching.  All  of  the  latter 
has  been  along  the  line  of  structural  design,  where  I  have 
been  using  the  work  of  the  mathematical  department.  In  the 
first  place,  in  my  experience  as  a  student,  mathematics  came 
fairly  easy  to  me.  I  found  that  when  an  examination  was 
imminent,  I  could  cram  up  for  it  the  night  before  and  forget 
it  afterwards.  That  is  about  what  nineteen  out  of  twenty 
students  will  do.  Complicated  notation  tends  to  discourage 
the  student  from  getting  what  is  extremely  important  to  get, 
namely,  fundamental  principles.  I  find  that  students  know 
their  mathematics  fairly  well  but  they  don't  know  how  to 
apply  it.  This,  it  seems  to  me,  is  far  more  important  for  them 
to  learn  than  such  extremely  complicated  mathematical  prob- 
lems as  are  often  given  them.  In  actual  engineering  experi- 
ence the  applications  of  any  but  these  fundamental  formulae 
are  few  and  far  apart.  In  the  very  infrequent  cases  where 
the  more  complicated  formulae  are  used  it  is  only  necessary  to 
refer  to  tables  in  the  text-books,  as  the  majority  of  successful 
engineers  do  today.  In  my  opinion,  the  ideal  engineer  need 
not  have  an  extremely  mathematical  training.  In  running  a 
railroad,  it  is  far  less  important  to  get  the  line  exactly  curved 
and  mathematically  accurate,  than  it  is  to  run  it  where  it  will 
cut  least  into  expenses,  which  is  the  main  point  involved. 
Imaginary  quantities  do  not  teach  this.  The  student  must  be 
taught  to  use  efficiency  engineering  in  handling  his  mathe- 
matics. If  this  can  be  taught  well,  we  shall  have  better  engi- 
neering students  than  if  we  attempt  to  teach  them  to  handle 
their  problems  by  imaginary  complex  quantities. 

Professor  G.  H.  Morse:  A  previous  speaker  has  referred 
to  alternating  currents  and  to  lack  of  familiarity  with  the 
mathematics  needed  for  this  subject.  I  wish  to  emphasize  the 
absolute  necessity  for  a  certain  amount  of  study  of  complex 
quantities  in  this  connection.     I  recently  made  a  tour  of  a 


DISCUSSION  ON  MATHEMATICS  REPORT.  131 

number  of  western  institutions — Illinois,  Purdue,  Armour 
Institute,  and  Wisconsin — with  the  object  of  discovering  how 
the  professors  were  teaching  electrical  engineering.  At  Illi- 
nois I  found  Professor  Berg,  who  spent  a  great  many  years  at 
the  General  Electric  works  developing  their  many  products. 
I  learned  that  he  has  given  up  entirely  all  methods  of  teaching 
alternating  currents  except  that  involving  the  use  of  complex 
quantities  illustrated  by  graphics,  of  course.  He  insists  upon 
this  method,  both  for  himself  and  his  assistants.  The  so-called 
trigonometrical  methods  have  no  standing  with  him  whatever. 
At  Purdue  I  found  Professor  Harding,  and  his  attitude,  while 
not  as  radical  as  that  of  Professor  Berg,  was  very  similar. 
In. my  own  case  I  find  that  the  use  of  complex  quantities  in 
teaching  alternating  currents  is  wonderfully  elucidating  in 
certain  parts  of  the  subject. 

Some  years  ago  I  had  the  notion  that  there  was  mathematics 
for  engineers  to  use,  the  kind  that  is  a  necessary  evil,  and  that 
there  was  mathematics  for  mathematicians,  in  which  they  had 
great  pleasure  in  soaring,  and  which  they  jealously  guarded 
from  use,  preferring  not  to  have  any  practical  applications 
made  of  it.  Since  I  have  been  associated  with  the  mathe- 
maticians at  the  University  of  Nebraska  my  ideas  have  entirely 
changed.  I  now  find  that  every  stage  of  these  fiights  in  pure 
mathematics  is  a  "short  cut."  The  higher  the  fiight  the 
shorter  and  more  useful  the  cut.  If  only  the  engineers  can 
appreciate  these  flights  their  work  will  be  greatly  simplified. 

Professor  S.  B.  Charters,  Jr.:  I  wish  to  emphasize  the 
fact  that  we  are  dealing  in  engineering  with  two  totally  differ- 
ent classes  of  students.  In  every  group,  in  the  proportion  of 
about  one  to  fifteen  or  twenty,  there  is  one  engineer.  Such  a 
man  should  have  and  will  take  and  enjoy  the  fullest  mathe- 
matical training.  On  the  other  hand,  the  comparatively  larger 
number  are  not  engineers  at  all.  They  are  simply  men  who 
are  getting  a  certain  amount  of  engineering  training;  and 
these  men  fill  the  bulk  of  the  positions.  From  the  colleges  of 
the  west  a  great  many  must  go  out  into  practical  work  as 
mining  superintendents,  superintendents  of  construction  in 


132  DISCUSSION  ON  MATHEMATICS  REPORT. 

the  installation  of  plants,  etc.  Now  that  class  of  work  absorbs 
the  bulk  of  our  men,  and  these  have  no  use  for  higher  mathe- 
matics whatever.  It  might  be  a  help  to  them  and  it  might  not. 
Among  those  graduates  whom  I  have  observed,  the  ones  who 
have  had  the  best  success  have  not  been  great  mathematicians. 
The  highest  paid  man  we  have  among  our  alumni  today,  is  one 
who  could  not  pass  any  mathematical  examination,  I  am 
reasonably  certain.  We  have  a  few  men  who  graduate  every 
year  who  should  be  given  higher  mathematics.  We  have  a 
feeling  that,  if  it  were  possible,  engineering  should  be  divided 
into  two  courses;  the  longer  course  of,  say,  five  or  six  years, 
with  adequate  mathematical  training,  for  the  man  who  shows 
special  aptitude  on  those  lines.  Those  men  should  be  the 
leaders  in  the  designing  branch  of  the  profession.  A  second 
class  of  men  need  not  have  the  higher  mathematics,  but  should 
have  the  proper  training  in  handling  men.  These  must  do  the 
bulk  of  the  work.  We  need  a  certain  number  of  men  to  do  the 
designing  and  hand  down  formulas  which  these  other  men  can 
follow.  We  need  more  men  to  take  those  mathematical 
formulae  and  from  them  get  the  results.  That  was  illustrated 
to  me  by  a  friend  who  stated  that  in  the  American  Bell  Tele- 
phone Company  there  is  one  man  who  does  the  principal 
mathematical  work  for  the  system.  In  each  division  they  have 
mathematicians  to  interpret  this  work  to  the  rank  and  file. 
Probably  twenty-five  or  thirty  experts  do  the  mathematical 
work  for  this  large  company  and  the  rest  of  it  is  done  by  the 
engineers  who  need  have  only  the  ordinary  mathematical 
training. 

Professor  H.  S.  Jacoby:  Allow  me  to  call  attention  to  the 
fact  that  this  report  deals  with  minimum  requirements,  and 
that  we  should  express  our  appreciation  of  the  splendid  work 
done  by  the  committee.  The  report  may  not  be  perfect  in 
every  part,  but  it  will  be  worth  a  great  deal  to  have  it  adopted, 
printed  and  made  available  to  the  teachers  whose  work  is 
affected  by  it.  It  may  be  made  a  starting  point  for  definite 
recommendations ;  changes  may  be  made  later  as  the  necessity 
for  them  appears.     If  in  any  institution  the  mathematical 


DISCUSSION  ON  MATHEMATICS  BEPORT.  133 

courses  are  of  such  a  character  as  to  require  enlargements  to 
conform  to  the  recommendations,  it  is  very  likely  that  they 
will  be  modified  in  time.  The  report  ought  not  to  be  a  hin- 
drance to  any  teacher  of  mathematics,  or  to  any  course  of 
study  which  is  now  more  extensive  in  its  scope. 

Professor  Webb:  It  occurs  to  me  that  there  is  something 
else  that  can  be  said  about  the  cause  of  the  trouble  between 
engineers  and  mathematicians.  An  engineer  very  often  has 
a  problem  that  he  does  not  see  through.  He  has  a  general 
idea  that  mathematics  is  a  powerful  instrument,  which  needs 
a  mathematician  to  solve  the  problem;  and  he  thinks  that  if 
he  knew  a  little  more  mathematics  he  could  solve  it  himself. 
As  a  matter  of  fact,  the  problem  may  be  very  simple  as  to  its 
mathematics,  and  it  may  be  only  that  he  does  not  see  through 
its  practical  or  engineering  side.  A  school  teacher  came  to 
me  with  a  problem  a  few  days  ago  and  said  she  had  given  it 
to  different  people  to  solv6,  and  some  advocated  one  solution 
and  some  another.  One  said  that  its  solution  needed  calculus ; 
I  thought  it  could  be  solved  quite  simply,  but  she  thought  not. 
This  was  a  problem  of  the  so-called  practical  variety.  A  barn 
forty  feet  square  has  a  horse  tethered  to  one  corner  of  it  by  a 
rope  one  hundred  feet  long.  How  much  grass  can  the  horse 
graze  over  without  going  over  the  same  grass  twice  ?  The  solu- 
tion of  this  is  very  simple,  but  one  should  not  expect  mathe- 
matics to  solve  it  before  the  problem  has  been  thoroughly 
analyzed.  Problems  of  this  nature  are  constantly  met  with  in 
engineering  work.  Very  little  mathematics  may  be  needed 
after  they  are  properly  analyzed,  but  if  this  calls  for  more 
common  engineering  sense  and  ingenuity  than  the  engineer 
has,  one  must  not  expect  the  average  mathematician,  much  less 
the  recruit  graduate,  to  make  good  the  deficiency. 


SYLLABUS   ON   COMPLEX   QUANTITIES* 

BY   CHAS.   O.    GUNTHER, 
Professor  of  Mathematics,  Stevens  Institute  of  Technology. 

1.  Derivation  of  formulae : 

e<^  =  cos^  +  tsin^,          e-<^  =  cos^  —  isin^, 
cos^:= ^ ,  tsin^  = ^ . 

2.  Definition  and  graphical  representation  of  a  complex 
quantity.  Polar  trigonometric  and  polar  exponential  equiva- 
lents of  z  =  x-{-  tt/,  that  is, 

2  =  it)(cos^  +  isin^),   polar  trigonometric; 
z  =  pe^^,  polar  exponential; 


in  which  p=V^^  +  V^  is  the  modulus  (the  positive  sign  being 
always  associated  with  it) ;  and  6,  given  by  the  relation 
tdJi6  =  y/Xy  is  the  argument  of  z.  Any  multiple  of  27r  may 
be  added  to  the  argument  without  altering  the  complex 
quantity. 

3.  Graphical  addition,  subtraction,  multiplication  and  divi- 
sion of  complex  quantities.  Graphical  solution  of  the  equa- 
tion x"  ±  1  =  0.    Logarithms  of  complex  quantities. 

Applications  to  Integration. 

4.  The  expression 

di/  =  tan2p^sec2»-+i^(i^, 
in  which  p  and  r  are  positive  integers  or  zero,  is  by  the  substi- 

•  This  syllabus  was  prepared  as  an  appendix  to  the  report  of  the 
Committee  on  the  Teaching  of  Mathematics  to  Engineering  Students  at 
the  request  of  the  members  of  the  Society  present  at  the  Pittsburgh 
meeting. 

134 


SYLLABUS   ON    COMPLEX   QUANTITIES.  135 

tution  tan  ^  =  t  sin  a  (a  being  an  imaginary  quantity)  trans- 
formed into 

dy  =  i{ — 1)  p  sin^^  a  cos^*"  ada. 

This  latter  expression  can  be  integrated  by  doubling  a  as  many 
times  as  necessary. 

The  foregoing  includes  the  integration  of  the  expression 

since  the  latter  expression  may  be  written 


dy 


=  _tan'.(-_.)sec-(|:-.).(^-.) 


As  found  in  text-books,  the  integration  of  the  expression 

dy  =  cos^esiBfede  (1) 

is  readily  accomplished  in  three  cases,  namely ; 

(a)  When  either  h  or  kis  an  odd  positive  integer. 

(6)  When  h-{-kis  a,n  even  negative  integer. 

(c)  When  both  h  and  k  are  even  positive  integers,  or  zero. 

The  first  two  of  these  cases  include  fractional  values  for 
h  and  k. 

By  means  of  the  substitution  given  above,  all  the  other  cases 
in  which  h  and  k  are  integers  can  be  brought  under  one  or 
more  of  the  three  cases  just  mentioned. 

In  the  above  are  also  included  all  the  cases  for  which  the 
general  binomial  differential  expression 

dy  =  x'^(a  +  &a;»)P/«dic 

can  be  integrated  without  resorting  to  infinite  series.  This 
expression  is  only  another  form  of  (1),  since  ^ya-\-hx**  can 
always  be  represented  by  one  of  the  three  sides  of  a  right 
triangle  and  therefore  expressed  as  a  trigonometric  function 
of  one  of  the  acute  angles  of  the  triangle. 

In  determining  the  value  of  a  definite  integral,  if  the 
variable  is  changed  the  limits  should  be  changed  to  correspond. 
For  example,  in  finding  the  length  of  the  arc  of  the  parabola, 


136  SYLLABUS   ON    COMPLEX   QUANTITIES. 

y^  =  4:ax,  from  the  vertex  to  the  point  (a,  2a),  we  have 


tan  9=1 


/»ian  9= 

=  2a   I 

•/tan  9=0 
fi  sin  a=l  /»t  sin  «=l 

==  2ai  I  cos^  ada  —  ai   \  (^  +  cos  2a)(Za 

*Ji  sin  a=0  «/i  sin  a=0 

Ji  sin  a=l 
i  Bin  «=0 

=  a[log.(i/2  +  l)  +  i/2]. 

Further  applications  of  complex  quantities  to  integration 
will  be  found  in  the  author's  discussion  on  p.  119. 


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